“Calling the stream you smuggles a self back in” assumes a self must be a substance. Buddhist philosophers, and Derek Parfit, argue that personal identity just is psychological continuity, with no further fact. Doesn’t the question assume the answer?

The Parfit comparison is apt; Parfit himself said his view was close to the Buddha’s. The Christian reply is that on Parfit’s view moral responsibility across time becomes a matter of degree and convention, which Parfit accepted and which most people find hard to accept, and that Buddhism’s karma doctrine seems to require more than convention. So the question is real, but it should be put precisely: can a no-further-fact view of persons support the full weight of karma? That is a question the tradition has noticed and argued about, not one it has missed.