Being a process doesn't preclude something from being an entity. In particular, all of us living organisms have our constituent atoms constantly replaced, and we don't typically consider that a risk to our sense of self.
> Being a process doesn't preclude something from being an entity
In a tortured semantics sense sure, but in a practical sense they refer to different types of things.
It comes down to "what stays the same when all its atoms are replaced" and "how can we tell it is the same entity and not just an identical but different one". I think the answers are "shape" AND "continuity".
E.g. all "atoms" of a glider in Conway's Life get replaced every couple turns but an observer can tell it is still a glider because it keeps the shape, and it the same glider because it continues it's previous state.
This makes AI not quite alive because it's missing the continuity.