I'd like to start giving people the option of commenting on shorter Cold Takes pieces (which I don't cross-post here or provide audio for). I'm going to use this post for that: I will generally leave a comment for each piece, and people can leave their comments as replies to that.
I've heard this view referred to as a time-slice view of personal identity before.
Personal identity is tied to ordinary questions about the identity and persistence of ordinary objects.
So, you should probably have the same set of persistence conditions (time-slice / constant replacement) for cups, computers, organisms, atoms etc.
If that's true, then "personality, relationships, and ongoing projects" are also only things that exist at a time-slices. Plausibly, they don't exist at all since each necessarily exists through time. Either way, there's no sense in which they can be shared with future selves.
I think this kind of issue is better solved by the "reductionist" understanding of Parfit's views than the "eliminativist" / "illusionist" version. There's no illusion of selfhood or constant replacement, just degrees of similarity that compose our idea of a self.