The Buddha's teachings are different from what Ajita Kesakambali taught.

The Buddha realized this (from [MN 19][1]):

> "When the mind was thus concentrated, purified, bright, unblemished,
> rid of defilement, pliant, malleable, steady, & attained to
> imperturbability, I directed it to the knowledge of recollecting my
> past lives. I recollected my manifold past lives, i.e., one birth,
> two... five, ten... fifty, a hundred, a thousand, a hundred thousand,
> many eons of cosmic contraction, many eons of cosmic expansion, many
> eons of cosmic contraction & expansion: 'There I had **such a name**,
> **belonged to such a clan**, had **such an appearance**. Such was my food,
> such my experience of pleasure & pain, **such the end of my life**.
> **Passing away from that state, I re-arose there**. There too I had such a
> name, belonged to such a clan, had such an appearance. Such was my
> food, such my experience of pleasure & pain, such the end of my life.
> Passing away from that state, I re-arose here.' Thus I remembered my
> manifold past lives in their modes & details. .....
> 
> "When the mind was thus concentrated, purified, bright, unblemished,
> rid of defilement, pliant, malleable, steady, & attained to
> imperturbability, I directed it to the knowledge of the passing away &
> reappearance of beings. I saw — by means of the divine eye, purified &
> surpassing the human — beings passing away & re-appearing, and I
> discerned how they are inferior & superior, beautiful & ugly,
> fortunate & unfortunate in accordance with their kamma: 'These beings
> — who were endowed with bad conduct of body, speech & mind, who
> reviled the 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
> the 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.'
> Thus — by means of the divine eye, purified & surpassing the human — **I
> saw beings passing away & re-appearing, and I discerned how they are
> inferior & superior, beautiful & ugly, fortunate & unfortunate in
> accordance with their kamma.**

The Buddha taught his followers to reflect in this way (from [AN 5.57][2]):

> “This noble disciple reflects thus: ‘I am not the only one who is the
> owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s
> origin, kamma as one’s relative, kamma as one’s resort; who will be
> the heir of whatever kamma, good or bad, that one does. All beings
> that come and go, that pass away and undergo rebirth, are owners of
> their kamma, heirs of their kamma; all have kamma as their origin,
> kamma as their relative, kamma as their resort; all will be heirs of
> whatever kamma, good or bad, that they do.’ As he often reflects on
> this theme, the path is generated. He pursues this path, develops it,
> and cultivates it. As he does so, the fetters are entirely abandoned
> and the underlying tendencies are uprooted.

From [SN 12.19][3]:

> “Bhikkhus, for the fool, hindered by ignorance and fettered by
> craving, this body has originated. For the fool that ignorance has not
> been abandoned and that craving has not been utterly destroyed. For
> what reason? Because the fool has not lived the holy life for the
> complete destruction of suffering. **Therefore, with the breakup of the
> body, the fool fares on to another body**. Faring on to another body, he
> is not freed from birth, aging, and death; not freed from sorrow,
> lamentation, pain, displeasure, and despair; not freed from suffering,
> I say.

From [SN 44.9][4]:

> “And, Master Gotama, when a being has laid down this body but has not
> yet been reborn in another body, what does Master Gotama declare to be
> its fuel on that occasion?”
> 
> “When, Vaccha, a being has laid down this body but has not yet been
> reborn in another body, I declare that it is fuelled by craving. For
> on that occasion craving is its fuel.”

Also, please note that the term "kāya" has been used in the sense of physical body for example in [SN 22.56][5] (although I know that it can be used to mean group or collection, when combined with other things):

> eye, ear, nose, tongue, body, and mind consciousness.    
> *cakkhuviññāṇaṃ, sotaviññāṇaṃ,ghānaviññāṇaṃ, jivhāviññāṇaṃ, **kāyaviññāṇaṃ**, manoviññāṇaṃ*

Pali words for body apart from "kāya" has been used in this context, for example "sarira" in [Dhammapada 400][6]:

> *Akkodhanam vatavantam      
> silavantam anussadam      
> dantam antimasariram<sup>1</sup>      
> tamaham brumi brahmanam*
> 
> Verse 400: Him I call a brahmana, who is free from anger, who
> practises austerity, who is virtuous and free from craving, who is
> controlled in his senses and for whom this body (i.e., existence) is
> the very last.
> 
> Footnote 1. antimasariram: lit., one who has the last body. This is
> his last body because he will not be reborn; he is an arahat.

From [SN 15.3][7]:

> "This is the greater: the tears you have shed while transmigrating &
> wandering this long, long time — crying & weeping from being joined
> with what is displeasing, being separated from what is pleasing — not
> the water in the four great oceans.
> 
> "Long have you (repeatedly) experienced the death of a mother. The
> tears you have shed over the death of a mother while transmigrating &
> wandering this long, long time — crying & weeping from being joined
> with what is displeasing, being separated from what is pleasing — are
> greater than the water in the four great oceans.
> 
> "Long have you (repeatedly) experienced the death of a father... the
> death of a brother... the death of a sister... the death of a son...
> the death of a daughter... loss with regard to relatives... loss with
> regard to wealth... loss with regard to disease. The tears you have
> shed over loss with regard to disease while transmigrating & wandering
> this long, long time — crying & weeping from being joined with what is
> displeasing, being separated from what is pleasing — are greater than
> the water in the four great oceans.
> 
> "Why is that? From an inconstruable beginning comes transmigration. A
> beginning point is not evident, though beings hindered by ignorance
> and fettered by craving are transmigrating & wandering on. Long have
> you thus experienced stress, experienced pain, experienced loss,
> swelling the cemeteries — enough to become disenchanted with all
> fabricated things, enough to become dispassionate, enough to be
> released."

Also, please note that since some people do not accept the Digha Nikaya as genuine teachings of the Buddha (please see [this question][8]), I have quoted from all the other nikayas instead. Dhammapada is part of KN.

  [1]: https://www.accesstoinsight.org/tipitaka/mn/mn.019.than.html
  [2]: https://suttacentral.net/an5.57/en/bodhi
  [3]: https://suttacentral.net/sn12.19/en/bodhi
  [4]: https://suttacentral.net/sn44.9/en/bodhi
  [5]: https://suttacentral.net/sn22.56/en/sujato
  [6]: http://www.tipitaka.net/tipitaka/dhp/verseload.php?verse=400
  [7]: https://www.accesstoinsight.org/tipitaka/sn/sn15/sn15.003.than.html
  [8]: https://buddhism.stackexchange.com/q/31099/471