This is an interesting topic. I think (hope) one day man will solve the mortality problem by uploading brains to computers (singularity) but until then the issue of mortality remains. Generic Subjective Continuity is a way to address this perhaps. But it needs a full blown theory of mind to be complete which (absent a theory of quantum gravity) eludes us right now. Some of the stuff here on this thread I need to address - first of all I never liked "b theory" (tenseless theory of time). It is a linguistic mistake - a category mistake. Temporality is the most basic thing there is, and it cannot be "waved away" as being (for example) a negatively curved dimension of anti-DeSitter spacetime. Those are just models, not reality as such. So, given "a theory" tensed time being the case, we need to ask "what is it" that is supposed to survive permanent brain death and "re-awake" in another nervous system in GSC? For me very roughly I think a conscious entity (as I have argued on my blog) is basically a mechanism that increases its internal entropy. So to be simple the "thing" that is conscious from moment to moment (or that awakens in the morning) is not "I" (the notion of "I" is a meme machine and that is another topic) , that pure "awareness" is not - in fact - a "thing" at all - it is a Process, a Process of entropy getting produced at an accelerating rate. (Tenseless time just muddles everything - with Wittgenstein I try to keep things simple where I can.) So the "thing" in GSC that "wakes up" after brain-death would be a Process that carries forth, in a new form. Consciousness is not "something that is" (a b theory red herring) but is a process that "is happening" - when I wake up, "consciousness is happening" or (in my view) "entropy is getting made". To channel quantum theory this thing we call consciousness is more a wave not a particle, it is an energy pattern (and / or informational pattern, if you will). Just as in quantum theory, electrons can be waves (which interact with one another and even recursively with themselves) - being simplistic here for clarity so I know i am oversimplyfing - so the process of entropy production we call consciousness is like a wave of information - it is not "I am concsious now" it is more like "a pattern of energy is conscious-ing" (verb). But to address another specious claim here - this notion of "we are all the same consciousness" (which would be horrifying if say I had to be re-incarnated as a victim of the Holocaust) is prima facia absurd - that would be like saying there is only one electron in the universe (again only possible due to b-theory b.s.). In good old fashion tensed , a-theory of time, the patterns of information that "conscious" (verb) can interact / overlap, sure, but are still also unique - we should think of "the soul" as a pattern of information / energy that generates entropy and it can have different contents (memories, etc) but is always unique , not reversable, not repeatable, etc. I think the quantum multiverse can explain GSC somewhat - if I die in one timeline, because "I" (the instance of awareness "I" associates with itself) is a wave (of sorts) it still "exists" in parallel timelines, ad eternam (or least ways until the heat death of the universe which perhaps is the Hindu Nirvana). Mind you, I still hope for / support the idea of technological immortality - I would personally rather live billions of years as a computer than die and get reborn and so forth but that is because the "I" is a meme-machine created by memes in order to produce memes and thus is naturally interested in its survival so it / "I" cannot here be objective.
Cool topic. (My own model of conscious entities as entropy-producing processes is here: theauguriesofamos[dot]blogspot[dot]com/2019/06/the-net-system-proposed-model-of[dot]html ).