Here's my thoughts on reincarnation, and as with all my beliefs they're always growing LOL!
Reincarnation of the body: Typically I don't really include this when I think about reincarnation, but you brought up biology and lineage so I thought I would, too.
Go back in time, all the way to the beginning of life on earth. There was one cell, or several types. One of these cells was the strongest, 'ate' the rest and began dividing. One cell became two, two became four, etc. Then it got more complicated, and these clusters divided and evolved to develop and specialize in the different lifeforms and processes we've come to know and understand. Ie - the circle of life. (I'll come back to that).
Now, the first humans. When two people (bears, tigers, fish) mate and conceive the first step comes from two cells: one from mom, one from dad. These start dividing the same way as that original cell, and poof! There's a new person who is really just an extension of the first. This person lives and dies, is buried in the earth and reenters the circle of life: decomposes, fertilizes the plants, plants energy is used to strengthen life so more can be made.
So, reincarnation of the body is just the use and reuse of cellular energy, though I might have some terms messed up. I'm an engineer not a biologist
Reincarnation of the soul: We are all connected, as I established before, biologically. But we are also connected spiritually. There is an energy that surrounds us all and flows through us, connecting us. This is what we use for magic and all those things that science and Christianity have a hard time explaining. But science has one thing right: Energy can neither be made nor destroyed. So, when we are born some of this energy is used to create our life. Not just the cellular processes, but our awareness, talents, personality. As we go through life we learn a bunch of stuff which strengthens (hopefully) that energy. When we die, where does that energy go? Back into the cycle. That energy that was your soul doesn't just dissipate, and it doesn't go to heaven or hell. That would imply that energy is constantly being taken from the earth and 'stored' somewhere, in a manner of speaking. No, it's needed to keep life going so it reenters the flow to be used again and again.
After death the soul doesn't automatically go back into a new body. It has time to go over what it has learned from the life it just ended and what it went through before. Some traditions call this the 'Summerlands', the place where your soul is at rest until needed again. When it is needed, it is used to 'power' the new body. The lessons learned are not entirely forgotten, but the ways to access them are. So we learn anew, but hopefully we can access things a little easier because we find them 'familiar' somehow. This makes it easier to learn even more, and more again until we've learned all the lessons there are to learn. I have no idea what happens then, because I personally don't think there is a way to learn everything
But I'll give it a shot!
I hope that helped and wasn't just a pointless rant because I haven't had enough coffee yet...
My experience in this life as a Norwegian/Welsh/ German American is only a small part of what I'm made of is another way of putting it) I believe I was a French woman in one lifetime and a British woman in another, but in another I could have been a male African warrior...or maybe for some reason I couldn't have been...I just wonder....