Souls being born into a body is a concept introduced in Christianity.
Sorry BigJoeNobody but that's not quite right.
As ever there is a necessity to qualify that because of the proliferation of denominations, but I would say outside of America, most if not all denominations follow the holistic Hebrew view of body-and-soul, rather than the dualist Hellenic view of a body-soul dichotomy.
As regards reincarnation in a Christian context, the dialogue is not an entirely closed book, but the
a priori problem is that most people assume that what reincarnates is the individual self ... and then the doctrine founders on contradictions, as far as I can see ... how many selves can a person have, and who is it, or what is it, that has all these selves?
This is prevalent in the Western mindset that holds a rather Hellenic notion of the 'soul' and moreover the 'self'. I don't think Hindus or Buddhists see the self quite so possessively as the West does, but the West tends not to discern what the original teachings are, but rather simply latches onto the idea ... mostly I suspect though FOMO!
If we take the soul as the essence of the self, then how can the soul be this self in this life, and yet another self in another life ... that's two selves, and unless we're assuming multiple personalities, the idea undermines the original concept of 'self' ... but that's as I see it.
If we are saying the soul is prior to the self, and indeed the self is just, as the Buddhists would say, the aggregate of temporal and ephemeral states, then I have grounds to discuss reincarnation ... but then that which people tell me reincarnates is not the self, but the aggregates, and the Buddhist at this point would shake his or head head and say no, that's not quite right.
Again with Hinduism. The soul can be said to be synonymous with Atman, and then one can discuss the soul-God relation in relation to the Atman-Brahman relation ... and furthermore I would say there is more congruence here in the two traditions, and indeed that
both traditions would shake their heads at the common and popular idea of reincarnation as, again, popular but somewhat erroneous because of the tendency towards 'self-possession' in its assumptions.