Reincarnation seems to imply (if not to entail) the eternal primacy of spirit life (unembodied, 'pure' spirit) above incarnation (embodiment).
That is; with reincarnation Men are primarily - first and last - spirit forms; and the history of a Man's being begins with being-a-spirit and ends with being-a-spirit.
This spirit undergoes a series of incarnations which may be (eventually, according to some versions of reincarnation) aiming-towards eternal status as a spirit. The spirit learns-from - or is otherwise affected-by - an incarnation; however, it is only the spirit which persists.
Or else there is an unending cycle of (re-) incarnations (and perhaps transformations, for instance to other beings such as animals) through-which the spirit moves serially. But, equally; only the spirit is eternal, with the multiple incarnate forms being left-behind.
But the resurrection of Jesus Christ, and his promise of resurrection to all who follow him, implies a final incarnation.
Resurrection (which is an eternal embodied state) is thus implicitly regarded as having a higher status than that of pure spirit.
For me this means that when someone becomes a Christian, he expects (or at least hopes) that his death will be followed by resurrection; and therefore Not by another reincarnation.
Against this understanding are ranged several of my spiritual mentors such as Rudolf Steiner and Owen Barfield, and William Arkle.
Steiner and Barfield are explicit that the ultimate and eternal aim of Man is to be a pure (discarnate) spirit; and that incarnations are 'merely' a series of 'descents' into the material, from-which the spirit is intended to learn.
For Steiner, Barfield and Arkle; the physical world, incarnation and bodies were obstacles that had certain advantages for learning particular life-lessons - but were ultimately intended to be dispensed-with.
William Arkle regarded multiple reincarnations over a vast stretch of history (some on earth, some perhaps elsewhere) as an essential element in the 'schooling' of the spirit, in the long and slow ascent of those who choose this path, towards full divinity.
My feeling has always been that Arkle was unclear about the role of Jesus Christ. He described ways in which Jesus was helpful; but did not provide any account of why Jesus was 'essential' to the ultimate goal of full divinity - or why the possibility of 'following' Jesus made a decisive difference.
My own understanding of the 'educative' function of this mortal life, which I developed-from Arkle's thinking rather than deriving it from what he explicitly said; is that our mortal life is indeed (as Arkle describes) as being like a university tailored around our specific personal needs.
But I see this life now as the chance to learn one or a few specific 'lessons' that are necessary to our full spiritual development. For example, we need to recognize and repent particular sins; or we need to love another person or persons, in order to be able to want to make that eternal commitment to live-by-love which is needed to want to enter Heaven.
Each person's actual (here-and-now) life is therefore being arranged by God for his or her very particular needs - needs which he brought into this life from his or her distinctive state of being as a pre-mortal spirit.
This is why I see no absolute need for a Christian to experience multiple incarnations. After Jesus made it possible for those who loved him to follow him (rather literally) through death and to resurrected eternal life in Heaven (as described in the Fourth Gospel), then multiple reincarnation was no longer needed - and was indeed impossible for the resurrected Man experiencing Life Everlasting.
I can only regard this combining of Christianity and reincarnation as an error - a metaphysical* incoherence. A failure correctly to discern and understand the core teaching of Jesus Christ and the demonstration that was his life, death, resurrection and ascension.
And I think the source of this error lies in the (common, almost universal) failure to regard the Fourth Gospel (termed 'John') as the primary Christian scripture; because this text seems to state quite explicitly (and repeatedly) that followers of Jesus can expect resurrection to life everlasting after the death of their mortal bodies: which clearly (so it seems) excludes the possibility of reincarnation.
A Christian who built his faith from the Fourth Gospel would (surely?) cease to expect - and to want - any further reincarnation?
*Note - By metaphysical incoherence I mean a matter of incompatible primary assumptions; which therefore has nothing to do with 'evidence' or 'observation', because metaphysical assumptions frame the nature and status of evidence and observation.
Further Note - I am not saying that there never was reincarnation; in the contrary I assume that it certainly has happened in some times and places, and possibly continues to happen. What I am saying is that following Jesus Christ to Heaven necessarily terminates the cycle of reincarnations.