The Dalai Lama gave a very good answer to this conundrum
In Buddhism, what we call as the 'self' designated as the combination of body and mind
At the subtle level, body and mind has no beginning and no end, thus it cannot 'die'
Upon death, subtle mind separates from the body and is what is carried over to the next ...
It's no different from your smartphone.
You have SMSes, chat history, photos, videos, notes etc.
Now let's say you have bought a new phone, so, you can manually or automatically (using a cloud-based service), transfer all the contents to a new phone, and sign in using your various accounts.
The phone doesn't need a soul or self for that.
Similarly, old karma ...