An answer to this question is in the Khajjanīya Sutta. The Khajjanīya Sutta explains life is comprised of five aggregates, namely: (i) body; (ii) feelings; (iii) perceptions; (iv) mental formations; & (v) sense consciousness. Mental stories about so-called or imaginary 'past lives' are 'mental formations', similar to dreams the mind constructs or manufactures in sleep.
The Khajjanīya Sutta states 'mental formations' about the past should be interpreted as follows:
Any kind of mental formations whatsoever … whether past, future or present, internal or external, gross or subtle, inferior or superior,
far or near, all mental formations should be seen as it really is with
correct wisdom thus: ‘This is not mine, this I am not, this is not my
As for 'rebirth', the Paticca-samuppada-vibhanga Sutta explains 'birth' ('jati') is the mental generation of the mental idea, view or 'assumption' of 'beings' ('satta').
SN 5.10 explains that, apart from 'view' or 'conceptual thought' (which are mental formations), there are no 'beings' to be found. SN 5.10 also states the idea of a 'being' (or 'soul') is a view held by Mara (Satan).
It follows, as explained in SN 22.85, each time the mind gives birth to the 'self' idea & believes it is a 'self', that is 'birth' ('jati'). To quote:
There is the case where an uninstructed, run-of-the-mill person — who has no regard for noble ones, is not well-versed or disciplined in
their Dhamma; who has no regard for men of integrity, is not
well-versed or disciplined in their Dhamma — assumes the body,
feeling, perception, mental fabrication &/or consciousness to be a
'self'. That assumption (of 'self') is a fabrication. Now what is the
cause, what is the origination, what is the birth (jati), what is the
coming-into-existence of that fabrication? To an uninstructed,
run-of-the-mill person, touched by that which is felt born of contact
with ignorance, craving arises. That fabrication (of 'self') is born
If the 'self' fabrication keeps arising in the mind, each new fabrication of the 'self' idea is a new 'becoming', a new birth or 'rebirth'.
Most importantly, regardless of how the words 'birth' ('jati') & the myriad terms translated as 'rebirth' are interpreted, the most salient feature of Buddhist rebirth is it is always connected to the results of kamma (actions), i.e., productive of happiness ('heaven') & unhappiness ('hell'). To quote a stock phrase:
'These beings — who were endowed with bad conduct of body, speech &
mind, who reviled noble ones, held wrong views and undertook actions
under the influence of wrong views — with the break-up of the body,
after death, have re-appeared in the plane of deprivation, the bad
destination, the lower realms, in hell. But these beings — who were
endowed with good conduct of body, speech, & mind, who did not revile
noble ones, who held right views and undertook actions under the
influence of right views — with the break-up of the body, after death,
have re-appeared in the good destinations, in the heavenly world.'
Unlike the Hindu reincarnation found in the Bhagavada Gita, which states the soul (atman) will always enter a new physical body at death, the idea of 'rebirth' in Buddhism is always connected to the results of kamma or moral efficiency.
Therefore, in Buddhism, unlike Hinduism & unlike the heretical branches of Judaism, 'rebirth' in Buddhism is not 'meta-physics' but, instead, 'morality' about results of kamma.
In other words, even Buddhists that do not believe in life after death still must believe in 'results of kamma' or 'moral rebirth' in order to have right view.
The language used in the Pali scriptures accommodates both views about 'rebirth' since terms such as 'beings' ('satta'), 'death' ('marana'), 'body' ('kaya'), etc, have dual meanings.
As previously stated & quoted, in the older Pali suttas, the predominant view about 'rebirth' is good kamma leads to 'heaven' & bad kamma leads to 'hell'. It seems only in later teachings did the idea of returning as a human being start.
In general, Buddhism teaches all things whatsoever are 'not-self' or, if using Hindu terminology, 'not-soul' ('anatta'). Therefore, Buddhism is contrary to Hindu & later-day-Jewish ideas about transmigration of souls. The Jewish Encyclopedia states:
'Transmigration' of souls (metempsychosis)...This doctrine was foreign to Judaism until about the eighth century, when, under the
influence of the Mohammedan mystics, it was adopted by the Karaites and other Jewish dissenters.
As for the pseudo-science of Ian Stevenson and Jim Tucker, since their claims arise from the studies of literally few children from billions, their claims can be debunked with explanations such as: (i) the child was conditioned by is parents; (ii) the child has a mental disposition or psychic power that can absorbs external ideas; and (iii) mystics with psychic powers are controlling the mind of the child.
For example, the failure of many so-called reincarnate Tibetan lamas, such as Tenzin Ösel Hita, shows such ideas about the reincarnation of souls are spurious or tenuous. In the case of Ösel Hita, it seemed to be not 'transmigration' but child abuse. To quote Ösel Hita:
"At 14 months I was recognized and taken to India. They dressed me in a yellow hat, they sat me on a throne, people worshipped me ... They
took me away from my family and put me in a medieval situation in
which I suffered a lot. It was like living a lie."
If we all were really reincarnated souls then we would all remember our past lives or we would all be born with knowledge & experience from our past lives. But, unfortunately, we are all born ignorant, like a blank slate with some natural instincts, and we all must learn how to live & survive in life, as though we all started out anew & fresh.
Some translators of old Buddhist scriptures use the word "transmigration" but this is just a mistranslation.
Similar to the Jewish dissenters, later-day Buddhists created many ideas about reincarnation & even ideas about how 'consciousness' (which is mere sense cognition) is reborn, which probably explains why Buddhism become extinct in India, since it became the same as Hinduism.
Even in the Pali suttas, there are maybe a dozen suttas from thousands (such as the Chariot Maker sutta) that state the Buddha had a literal past life. These suttas are questionable as to their authenticity given they contradict the Khajjanīya Sutta.
In summary, original Buddhism taught the idea of 'self' is merely a mental formation/fabrication born from ignorance and that, in ultimate reality, there is no 'self' to be found. Therefore, the idea that 'my soul transmigrated' or the idea that "I" had a past life is contrary to original Buddhism.