How do Buddhists who believe in a non momentary conventional self during life account for karma without enlightenment?

How do Buddhists who believe in a non momentary conventional self during life account for karma without enlightenment? Specifically, what happens to "my" karma when I die, if I am not enlightened and my karma has not been exhausted? I'm not asking how to think of cause and result if the lack of self identity extends from moment to moment in this life, because then "my" karma would face a similar obstacle in this life (there being no personal identity from moment to moment means it cannot be "my" karma at all). Clearly most historical Buddhists believed that karma did not just exist in the suffering I cause others, but that what comes around goes around. So if my life does not, automatically without enlightenment, generate a new one, why doesn't "my" death cut off that cycle?