This is somewhat disturbing to think about, the current you will no longer exist... Anyway, feelings aside:

What do you define as the current? Where do you draw the line of this moment and the following? Do you die thousands of times a second, every 5 seconds? Can you put a duration to the current "you"?

Memories are interesting. Coming from a tech background, I like to think of memories as indexed moments of time. Just as databases exist to be efficient in retrieval and storage, memories tend to work very well when they're also associated with something more than the memory itself.

I don't know if I'm doing a good job explaining anything, so please ask whatever that's needed!..

Anyway, building a little more off the memories as an indexes in a database - would you argue that the dying entities include entities that are working to constantly maintain and index these memories? Do they even get stored, or does the next self simply know all of the previous? Does this same self exist subconsciously with everything the body requires to survive (knowing when to breathe, feeding air to the body, nutrients from food, creating blood, basically every biological)?