Reincarnation is a metaphor. As explained by the Buddha, Nagarjuna, and various other Buddhist sages and philosophers (as well, we might add, by various Western philosophers), the Self is a construct that has no intrinsic characteristics or self-supported “reality.” It is a construct assembled out of sense impressions and, in a Buddhist sense, karmic seeds, from moment to moment. From a two truths perspective, the conception of the Self is conventionally real in that it helps navigation through life, but it is no more real than any other tangible or intangible object. Ultimately, the Self is empty, just like all other observed objects and phenomena.
Given this, that the Self is empty just like everything else, how could there be anything that continues “from life to life”? There can’t be such a thing or such a continuation. Reincarnation, then, is best thought of as another description of the dependently-arising object that is reassembled from moment to moment from sense inputs received and karmic seeds retained. Alternatively, perhaps there is no need to think about any concept of reincarnation at all – it is easily discarded if it is not helpful in one’s practice!
Reincarnation requires there to be a permanent Self, making permanent changes, in form and place. From an emptiness perspective that’s simply not possible.