I'm tempted to bring up the concept of dependent origination from Buddhism, but this thread is already messed enough as it is.
Oh no! I just did! What's that swirling vortex of darkness coming down from the sky!?
You, the thread owner, need to think more deeply about what it means to be a me, or a self, and whether that concept makes sense in all the relevant contexts. (The ship of Theseus paradox has already been touched on, but that's a good start. The Buddhists have a concept, anatman, and the doctrine of Anatta, in which there is no enduring "self thing" to be recreated; examining Buddhist doctrine may not provide answers, but Buddhist defenses of Anatta, say on arguments about the five skhandas is a good place to look for missteps that have already been outlined.) I specialize in philosophy of mind, and from what my limited exploration has uncovered, there is no well founded theory of what constitutes a self. (Not including my own theory, which may or mar not be well founded.)
So it's not even clear that there is even a coherent concept of 'self' to which the process of reincarnation would apply, or at least not one that's well known.