Buddhists do not believe in reincarnation, the view is rebirth.
Try this. According to the TV show Madmen, there is a Japanese saying that says that a man is whatever room he is standing in. So in a business office, he IS a businessman, at home, he is a husband and a father, at a bar, he is a drunk, etc.
Now a man can walk from room to room, showing a continuity from "life" to "life". Clearly there is some sort of continuity, but when the man was in the business office, he was not the drunk, and vise versa, so we have no true business man, no true drunk. Just the appearance of them.
Rebirth says the man really is whatever room that he is standing in, and as that man he is as real as that room, and this is possible precisely because there is no true, central core man behind the causes and conditions that give rise to the specific appearance of the business man. If there were a non-empty self, it would not be subject to causality, and would not, therefore be capable of taking on an appearance, or having interactions. The appearance of a self is all there ever is, and it is precisely the clinging and grasping onto the basic notion of a true self at all that perpetuates rebirth from room to room.
To understand rebirth a bit more clearly, we can realize that at no point did we ever stop being a child, and start being an adult, but looking back, it is clear that one is different from the other. In terms of lives, we cannot say when one definitively ends and another begins, we can merely say that we can point to the apparent beginning and end based off of certain inferential and ultimately arbitrary definitions. There is never a clear moment when one room ends and another begins, never a clear moment when the businessman starts, and the businessman stops. We just "draw a line".
To go a bit further, because we cannot assert a true beginning, there is no way to assert a true lifetime to reincarnate into, there is merely the codependent origination of apparent phenomena. The whole thing is nothing but the perpetuated arising of empty phenomena due to causes and conditions giving rise to other causes and conditions, but nothing ever truly breaks over into actual true existence. We seem to perceive this, but that is just a relativistic assumption.
TL;DR - Buddhists do not believe in reincarnation or true separate lifetimes, so there is no problem, there is merely the perpetuation of the causes and conditions that give rise to apparent, yet ultimately empty phenomena.