I lean towards a sort of death. But perhaps we're all dying and reanimating in between each Planck time instant anyway, in which case we shouldn't worry about it too much :)
But in terms of selves, of consciousness, they're all isolated phenonema. I think consciousness is merely an illusion, we think we have continuity because we inherit the memories and thought processes from the self from the moment before, but all that exists is a moment of awareness (the "now"), which is analogous to a clock of a computer processor.
The upside, of course, is that my philosophy frees me up for gultfree teletransportation. Bring it on!
Physically¹ events happen instantaneously we just can't tell at which precise instant they happen. This is true even with quantised time.
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¹ Perhaps that should say "in the mathematical model of the physical universe we base our science on".