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 perhaps 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.
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.