This is all highly speculative, but I think that looking for a material basis of rebirth is betting on the wrong horse. I would definitely bet on a nonphysical horse here. At this particular stage in history, the success of science directs attention away from the nonphysical, since science itself is embedded in a materialist/physicalist ontology. I would look at epigenetic processes before looking at heredity.
Apart from that, I think it is -at least in principle- possible that empirical sciences can shed some light on rebirth. The study conducted by Ian Stevenson "Twenty cases suggestive of reincarnation" has pioneered this field, and this work is now being continued from various sides by various researchers. I think it is somewhat comparable to the phenomenon of NDEs (near-death experiences). 30-40 years ago, NDEs were a rather obscure phenomenon and almost no literature existed on the topic. Today we have roughly a dozen scientific studies on NDEs that were published in respected science publications and we have a flood of books that have popularised the topic. The research on PLEs (past life experiences) will not be as easy as the research on NDE, because the phenomenon is much rarer and presents intricate internal problems for a scientific point of view, but I think we can expect an increasing body of literature and empirical support.