This question was brought to the fore for me by my recent engagement with William Arkle and by a much more superficial recent overview of the life and works of Rudolf Steiner - both of whom (despite their very obvious heterodoxies and heresies) saw themselves as Christian, and both of whom (to all appearances) seem genuinely inspired and enlightened (albeit imperfect) men, and also good men.
And both believed in reincarnation - partly from what they perceived to be logical necessity, but also from what they believed to be divine revelations.
There are also many other - and much more famous and influential - religious figures who put great emphasis on reincarnation. The founders of Hinduism and Buddhism to name only the two largest; but probably most humans who have ever lived believed in some version of reincarnation, as well as many or most of the greatest religious founders and leaders. .
Given that reincarnation is not a part of core Christian revelation - and seems hardly likely to have been forgotten or left-out by mistake - then how to account for this belief?
Perhaps the most frequent explanation throughout Christian history would be that these are demonic deceptions, designed to manipulate mankind into a state of unrepented sin. But looking at the big picture of these religious figures and of the vast mass of their adherents, makes this seem implausible.
One possibility is that the religious leaders, who believe that they have been given a revelation of reincarnation, are correct: but only with respect to themselves.
In other words, reincarnation is a possibility - with some examples being given in the Bible - but it sees to be a rare exception rather than the norm or the rule.
In other words, perhaps these inspired spiritual men - such as Steiner and Arkle - are actually themselves among the rare instances of reincarnation; or else are destined to be?
Perhaps they recognise this fact, but mistakenly (but not maliciously) extrapolate it to the mass of other people for whom mortal life is a one-off - and teach it to these people (for whom reincarnation is not, in fact, true) - and these people often accept it due to the genuine spiritual authority of the teachers.
For instance, if someone is himself a reincarnate, then it may be one reason for his advanced spiritual state - he has had more than one life-time to progress, a greater experience; and has consented to return and teach from this enlarged experience.
So these spiritual teachers are indeed (perhaps) reincarnated souls; and they have a particular reason for being reincarnated; but once re-born they are subject to the usual constraints of mortal earthly life; so, although spiritually-advanced, they remain imperfect and prone to both error and sin - including the error (but not necessarily sin) of over-generalising from their own case.
(Indeed, the over-generalising error may plausibly result from humility - the reincarnate perhaps cannot believe his almost unique status and elevated destiny.)
Anyway, for what it is worth, this struck me as a possible - and in some ways satisfying - explanation for the recurrence of the reincarnation idea in a Christian context.
Note added: Not all the people who believe in and/or teach reincarnation, not even those who give evidence of genuine inspiration, will necessarily themselves actually be a reincarnate. It is quite possible that the majority are not reincarnated nor will become reincarnated. All I am saying is that it is (I think) a possibility that some teachers of reincarnation may be telling the truth (within the usual constraints of human imperfection of understanding and expression) - so far as they personally are concerned.