The vanished moment is hypochondria. It cannot pass or upload anything to the future although it still is able to cause new moment effects.

The past vanished moment are not upload anything to its effect moments. It just causes its effect to arise. However, the effects are able to arise as the same habit, e.g. same object and same feeling, because of Upanissaya Paccaya.
The vanished moment is hypochondria. It cannot pass or upload anything to the future although it still is able to cause new moment effects.
For example, I can smile when I thinking of my dead sister however she vanished and doesn't upload anything to me, she just causes me to smile.

Rebirth appears in both sutta and abhidhamma (Paṭisandhi, Okkamati, Jāti); Fear of Rebirth; Sutta-Connection

Related topic: https://unmixedtheravada.blogspot.com/2019/01/consciousness-in-mn-49-dn-11-is-wrong.html

1. MN 38 of Sāti-bhikkhu is about the single consciousness traveling, viññāna-sakkāya-diṭṭhi. It never arises or vanishes, independent form the origination, and can control itself through the new body/life.
   But Paṭisandhi of Abhidhamma is about many various aggregates conditioning each other by many methods.
   

   For the example, a human rebirth has 33 mental factors+1 consciousnesses+30 matters are conascence depending on each other. They all depending on their past karmma mental factor, which depending on its 33 conascence mental factors+1 consciousness+many matters (rebirth from matter-realm). All those consciousnesses, mental factors, and matters also are depending on their uncountable previous arisen consciousnesses, mental factors, and matters. This is just a brief example. There is millions more explanation in 12 briefed abhidhamma-piṭaka-canons. It still is only little Leaves in Buddha's hand

   By the above example, you can see Paṭisandhi, it's conditioners, and it's conditioned things are depending on many other conditioners. They all are not independent, cannot control itself, so it is completely different from MN 38.
   The main idea of Vipassanā Meditation is the practitioner can realize impermanent, suffering, and non-self more and more when the practitioner analyzes aggregates' conditioners more and more in detail.
2. Paṭisandhi, the re-linking, is a term of bh.Sāriputta, the great follower, it often appears in every bh.Sāriputta's canon because he is the representative of the Buddha at Buddha-living-time in AN Ekakanipāta Ekapuggalavagga, so it is his job to explain Buddha's teaching. It appeared in MN Mahāgosiṅgasālasutta that most of the foremost Buddha's followers go listen to Sāriputta's sermon.
3. Buddha uses many various other words which mean "rebirth" instead of bh. Sāriputta's "Paṭisandhi". See my answer in this question, and notice the words using of quoted sutta in that answer, it is using in the next.
4. Bh.Sāriputta defined "Paṭisandhi" from DN 15 Mahāvagga Mahānidānasutta:
   

   Name-&-Form
   “‘From consciousness as a requisite condition comes name-&-form.’ Thus it has been said. And this is the way to understand how from consciousness as a requisite condition comes name-&-form. If consciousness were not to descend into the mother’s womb, would name-&-form take shape in the womb?”
   “No, lord.”
   “If, after descending into the womb, consciousness were to depart, would name-&-form be produced for this world?”
   “No, lord.”
   “If the consciousness of the young boy or girl were to be cut off, would name-&-form ripen, grow, and reach maturity?”
   “No, lord.”
   “Thus this is a cause, this is a reason, this is an origination, this is a requisite condition for name-&-form, i.e., consciousness.
   Consciousness
   “From name-&-form as a requisite condition comes consciousness.’ Thus it has been said. And this is the way to understand how from name-&-form as a requisite condition comes consciousness. If consciousness were not to gain a foothold in name-&-form, would a coming-into-play of the origination of birth, aging, death, and stress in the future be discerned?”
   “No, lord.”
   “Thus this is a cause, this is a reason, this is an origination, this is a requisite condition for consciousness, i.e., name-&-form.
   “This is the extent to which there is birth, aging, death, passing away, and re-arising. This is the extent to which there are means of designation, expression, and delineation. This is the extent to which the dimension of discernment extends, the extent to which the cycle revolves for the manifesting [discernibility] of this world—i.e., name-&-form together with consciousness.

5. Buddha didn't answer the ten questions of bh. Māluṅkya in MN CūḷaMāluṅkyOvādasuttaṃ because bh. Māluṅkya, young, was caving four gains at that time. The story appears in MN MahāMāluṅkyOvādasuttaṃ, AN Māluṅkayaputtasuttaṃ when bh.Maluṅkaya got old, KN Thera. MaluṅkayaputtaTheraGathā after old bh.Māluṅkaya enlightened, and you can see in their Atthakathā as well.
   However actually, Buddha already answered it in DN1 Brahmajālasuttaṃ, and Buddha described the dependent origination part of DN1 in DN 15 Mahāvagga Mahānidānasutta quoted above.
6. Even Tipitaka and Atthakathā are written on papers as the public domain license, they still recite&memorize orally in Mahāvihāra Theravāda Buddhist Line, such as Pa-Auk. We never deny people to comment on Tipitaka and Atthakathā, but it's Mahāvihāra Tipiṭaka Memorizer Right to adjust the tipiṭaka users' misunderstood because we have lesser of the lacking in reading because through 2600 years we still recite&memorize Tipitaka and Atthakathā, not just read through.
   The similitude is the chemist who has recited, memorized, and learned about the periodic table and the formulas from the past chemist generation to generation.
7. If one still want to see their past life, your welcome to Pa-Auk monastery. Here you can practice to recalling your past life as long as you can, but you must meditate Jhāna before the recalling step.
8. Actually, abhidhammic people not often talk about the rebirth too much in our daily life because we already have known of the rebirth in detail. People Fear What They Don't Understand, so abhidhammic people have no fear in rebirth either there is the real rebirth or not. We don't debate people to protect "rebirth theory" and "no rebirth theory" because we are not eternalist and nihilism. We just explain people's misunderstood of "rebirth" in tipitaka-pali to keep Tipitaka goes on till 5000 years because the "no rebirth theory" supporters are trying to cut many sutta off Tipiṭaka. This is dangerous equal to the "rebirth theory" supporters try to add the new sutta into Tipiṭaka.
9. Are the "no rebirth theory" supporters are fearing of the rebirth? Why they often talk about the rebirth without the understood of it in detail? Is that ignorant of the rebirth information leading them to be fear of the rebirth? Is it the main cause that they try to cut many sutta about heaven and hell off Tipiṭaka? Is it called doubt-fetter in SN12.20 Paccayasutta?
   

   "When a disciple of the noble ones has seen well with right discernment this dependent co-arising & these dependently co-arisen phenomena as they have come to be, it is not possible that he would run after the past, thinking, 'Was I in the past? Was I not in the past? What was I in the past? How was I in the past? Having been what, what was I in the past?' or that he would run after the future, thinking, 'Shall I be in the future? Shall I not be in the future? What shall I be in the future? How shall I be in the future? Having been what, what shall I be in the future?' or that he would be inwardly perplexed about the immediate present, thinking, 'Am I? Am I not? What am I? How am I? Where has this being come from? Where is it bound?' Such a thing is not possible. Why is that? Because the disciple of the noble ones has seen well with right discernment this dependent co-arising & these dependently co-arisen phenomena as they have come to be."

In AN 4.171, what is “reincarnation where the intention of others is effective”?

Per commentary, this passage refers to "devas corrupted by the mind" (Manopadosika-deva), living in the world of the Four Great Kings. When they get angry with one another their mutual anger leads to their mutual deaths.
Here are descriptions of Manopadosika-deva from different sources:
Buddhist Dictionary. Manual of Buddhist Terms & Doctrines (4th Edition) by Ven. Nyanatiloka:

Manopadosika-deva: ‘the celestial beings corruptible by temper’, are a class of devas (q.v.) of the sensuous sphere. “They spend their time in becoming annoyed with one another, and getting into a temper, and thus by being bodily and mentally exhausted, they pass from that world” (D. 1; 24).

The "Dictionary of Pali Proper Names" by G. P. Malalasekera:

A class of devas living in the Cātummahārājika world (DA.i.114; AA.ii.544; MNid.108). They burn continually with envy one against another (Buddhaghosa relates a story illustrating this in DA.i.114) and their hearts become ill disposed and debauched. Their bodies thus become feeble and their minds imbecile, and, as a consequence, they fall from their state (D.i.20; iii.32). The Manopadosikā were among those present at the preaching of the Mahāsamaya Sutta (D.ii.260). They are so called because their minds are debauched by envy (dosassa anudahanatāya) (VibhA.498).

Brahmajāla Sutta (DN 1) translated from the Pali by Bhikkhu Bodhi:

There are, bhikkhus, certain gods called 'corrupted by mind.' These gods contemplate one another with excessive envy. As a consequence their minds becomes corrupted by anger towards one another. When their minds are corrupted by anger, their bodies and minds become exhausted and consequently, they pass away from that plane.

Brahmajāla Sutta (DN 1) translated by Bhikkhu Sujato:

There are gods named ‘malevolent’. They spend too much time gazing at each other, so they grow angry with each other, and their bodies and minds get tired. They pass away from that group of gods.

So, one deva's hatred towards another deva provokes hatred in that deva towards the first, causing a vicious circle, and this very hatred exhausts their spirits until they fall from their godly state.
One deva's hatred is not his own deliberate intention but is merely a reaction to another deva's hatred. Therefore the deva falls from his state and comes to a different state through an "intention of other".
It's interesting how the vicious circle of judgement starts not through one of the devas performing some unwelcomed act to another, but rather through both spending too much time wondering about the other's attitude to oneself.
I suppose this basically means that not all karma is strictly personal. While in the simplest cases sentient beings inherit their own karma, there are cases when they create (bad) karma for one another, and there are cases when a karmic pattern emerges spontaneously from a background of ignorance - not through anyone's fault.
On a side note, per my (Russian) dictionary, attabhāvapaṭilābhe is not "reincarnation" but "a state of personal existence".
by Andrei Volkov @https://buddhism.stackexchange.com/a/30153/10100.

---------------------

I agree with Andrei Volkov explanation, which follows to it's atthakathā. However, for the perfect understood by sutta, not only atthakathā, I write this answer.

1. There is a reincarnation where one’s own intention is effective, not that of others. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe attasañcetanā kamati, no parasañcetanā.

By atthakathā, above text refers to DN Brahmajālasuttaṃ:

7. ‘And what is the second?

‘There are, brethren, certain gods called the “Debauched by Pleasure” [155]. ‘For ages they pass their time in the pursuit of the laughter and sport of sensual lusts. In consequence thereof their self-possession is corrupted, and through the loss of their self-control they fall from that state [156].

8. ‘Now it might well be, brethren, that some being, on his falling from that state, should come hither. And having come hither he should, as in the last case, become a recluse, and acquire the power of recollecting his last birth, but only his last one.
9. ‘And he would say to himself: “Those gods who are not debauched by pleasure are stedfast, immutable, eternal, of a nature that knows no change, and they will remain so forever and ever. [20] But we-who fell from that state, having lost our self-control through being debauched by pleasure-we have come hither as being impermanent, mutable, limited in duration of life.”

By atthakathā, above text refers to MN kandarakasuttaṃ:

2. There is a reincarnation where the intention of others is effective, not one’s own. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe parasañcetanā kamati, no attasañcetanā.

Bhikkhus, which person torments others yoked to it? Bhikkhus, here a certain person is a killer of pigs, goats, birds, a hunter of wild animals, a fisherman, a robber, a highway robber, one who keeps a prison, or engaged in any other vicious activity. This person torments others yoked to tormenting others.

3. There is a reincarnation where both one’s own and others’ intentions are effective. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe attasañcetanā ca kamati parasañcetanā ca.

By atthakathā, above text refers to DN Brahmajālasuttaṃ:

10. ‘And what is the third?

‘There are, brethren, certain gods called "the Debauched in Mind [157].” They burn continually with envy [158] one against another, and being thus irritated, their hearts become ill-disposed towards each other, and being thus debauched, their bodies become feeble, and their minds imbecile. And those gods fall from that state.

11. ‘Now it might well be, brethren, that some [\q 034/] being, on his falling from that state, should come hither; and having become a recluse should ‘as in the other cases, acquire the power of recollecting his last birth, but only his last one.
12. ‘And lie would say to himself: “Those gods who are not debauched in mind do not continually burn with envy against each other, so their hearts do not become evil disposed one towards another, nor their bodies feeble and their minds imbecile. Therefore they fall not from that state; they are stedfast, immutable, eternal, of a nature that knows no change, and they will remain so forever and ever. [21] But we were corrupted in mind, being constantly excited by envy against one another. And being thus envious and corrupt our bodies became feeble, and our minds imbecile, and we fell from that state, and have come hither as Being impermanent, mutable, limited in duration of life.”

‘This, brethren, is the third case.

4. There is a reincarnation where neither one’s own nor others’ intentions are effective. Atthi, bhikkhave, attabhāvapaṭilābho, yasmiṃ attabhāvapaṭilābhe nevattasañcetanā kamati, no parasañcetanā.

By atthakathā, above text refers to DN Dānamahapphalasutta:

"Having given this gift seeking his own profit — with a mind attached [to the reward], seeking to store up for himself, [with the thought], 'I'll enjoy this after death' — on the break-up of the body, after death, he reappears in the company of the Devas of the Four Great Kings. Then, having exhausted that action, that power, that status, that sovereignty, he is a returner, coming back to this world.
"Having given this gift with the thought, 'Giving is good,' on the break-up of the body, after death, he reappears in the company of the Devas of the Thirty-three. Then, having exhausted that action, that power, that status, that sovereignty, he is a returner, coming back to this world.
...

Why I have to link? Theravāda tradition never allow to explain tipitaka without reference, linking. It's weird and not professional. If buddhaghosa wrote something without the reference, he wouldn't be allowed to translate atthakathā back to pāli. So, the link, which I have given, show tipitaka memorizer's thought.

Do beings have control where they get re-incarnated

It is possible, however it is off chance.
How about that off chance possible?
The reincarnator must does:

1. Much enough merit.
   Because the merits cause the rebirth in a new good life, such as a human life, angel.
   See: K.N.Dhammapada's Commentary Patipūjikā, and chaddanta-jātaka's commentary.
2. Very strong efforts of mind's desire, intention, management, inspiration, decision, etc, to act about "specified new life" in every life's moments, until he die.
   Because the strong efforts to keep regularly actions that cause of target new life, will send you to be the specified new life, as a director. It is like a transgender man who use every second of life to act like a woman.
   See: A.N. Sattakanipāta Saṃyogasuttaṃ.
3. Die just in the same millisecond of new life fertilization.
   Because the dead-mind is immediate contiguity cause of rebirth-mind. And mind arising and vanishing is very fast (over trillions times per snap finger).
   See: Sanṅgaha 4th chapter.

However it is still very hard, so buddha taught in Pansu Suttas:

"The great earth is far greater, lord. The little bit of dust the Blessed One has picked up with the tip of his fingernail is next to nothing. It doesn't even count. It's no comparison. It's not even a fraction, this little bit of dust the Blessed One has picked up with the tip of his fingernail, when compared with the great earth.
"In the same way, monks, few are the beings who, on passing away from the human realm, are reborn among human beings. Far more are the beings who, on passing away from the human realm, are reborn in the animal womb... in the domain of the hungry ghosts.

You can see that it is very very hard to success. And all the way, you still will be old and dead after that. So, theravāda buddhism never support whole rebirth and reincarnation. We have jātaka, rebirth cannon, just for an inspiration and faith in buddha's ability and experience.

Nowhere you can avoid the suffering, if you still rebirth

This image captured from TEACHING & TRAINING book of Pa-auk monastery:

Remembering past lives

For adults

For adults, they can not remember because it is very far over you can remember it.

Take it easy. Don't look too far, just in this life, you have forgoten everything in your 1-2 years old period already.

For babies

For babies, Take it easy. Just see the amnesia-patient, it is the same but the reborn is worse than amnesia because reborn-human has an amnesia which is caused by loosen brain, not just damage, also has an amnesia which is caused by psychological trauma, which is caused bad environment in mother's stomach, too. So the babies loose whole of their past life memories.

In very advance theory, abhidhamma, babies can not remember because all of their memory-organs [saññāya āyatana] have change to very weak and none ability, similar to the old paralysis-man's organs. Also, they have to live in stressful stomach that has 37 °C temperature for at least 7 months, without vision. This is called kammajja-vāta [moving/changing form that arise by kamma] in suttanta-pāli tradition.

Kammajja-vāta is one element of kammajja-rūpa [form that arise by kamma]. All organs arising at the beginning moment of life, especially human organs, are very weak, loose every ability, walk/see/listen/etc, from the past life.

Mind and it's factors, such as saññā (memory factor), arise depend on those kammajja-rūpa which is a elements of purejāta-paccaya (pre arising cause) in paṭṭhāna, the 7th canon of abhidhamma. If this paccaya, sense-organs, so weak, saññā that arise at these sense-organs also weak too. Then saññā cause of sati (mindfulness, recalling ability mind's factor) by the definition "sati thirasaññāpadaṭṭānā". So the babies loose whole of their past life memories.

Theravāda recalling skill practice

In theravāda tradition, you can recall it by jhāna-based ñāṇa, such as pupbenivāsānussati-ñāṇa or paccayapariggaha-ñāṇa. This tradition still going on at pa-auk monastery and some Burmese monastery.

Self is paññatti. It is not grasping. Also, we choose to be mahāvihāra theravāda buddhist because we want to get all security.

Self is paññatti. It is not grasping. Grasping arise to attach/to cling khandha and paññatti, that are grasping's object. But paññatti never arise, so self never begin or end. Similarly, when you dream. The dreaming-mind is real and arising. But it's object never arise and never vanish, just an imagination of mind. So, self is not real.
Furthermore, for aggregates and grasping, in next life, is already taught that still arise until an ignorant and a craving still is not destroyed by arahanta's enlightenment . Because, there is a future-clinging-aggregate, and that clinging will try (kamma-bhava, kamma-becoming) to have (before death=get new thing, after death=rebirth) that future-clinging-aggregate. So in Khandhasutta taught:

"Whatever form — past, future (form-clinging-aggregate), or present; internal or external; blatant or subtle; common or sublime; far or near — is clingable, offers sustenance, and is accompanied with mental fermentation: That is called the form clinging-aggregate.

And Paṭiccasamuppāda taught:

From craving as a requisite condition comes clinging/sustenance. From clinging/sustenance as a requisite condition comes becoming. From becoming as a requisite condition comes birth.

Furthermore, we choose to be mahāvihāra theravāda buddhist (because we using mahāvihāra's tipitaka) because we want to have anytime and anywhere security by make every possible protection to protect every of your future self from every possible insecure. If you still lazy to protect your self from next life/rebirth (by making sure you [five aggregates] must never arise anymore), you are not mahāvihāra theravāda buddhist practitioner. Also, you never understand throughout the theravāda-tipitaka's concept.

I'm wondering WHY a strong belief in rebirth does not conform with the above Buddha's teachings?

Because in the western history, the western people have suffering by some religious, that have a strong belief in rebirth through moreover 2000 years, so they hate that religious and that belief, together. when they catch the buddha's teaching, they bias the buddha's teaching by that hate. They try to cutting rebirth off tipitaka, although they never completely study throughout the buddha's teaching by an ancient study system