Now lets deal with the darned continuity ...
One problem. You're not dealing with continuity of the self.
scenario #1:
You are a healthy but skinny, nice young person living at location A-town (let's call this state A). After a while you become dissatisfied with your appearance. Your start to change your body. Cosmetic surgery is cheap in A-town, so you get yourself a new face, new hair, start to build up muscle etc. (state B). You move to C-town at the other side of the planet (state C). As you get older you slowly but surely become an old grumpy bastard.

Nothing is left of your once nice personality (state D). At old age, you catch Alzheimer disease and slowly degenerate. After a few years with the disease, you have totally forgotten your friends, family and how it was to live in A-town 70 years earlier (state E).
Transitions A to B, B to C, and C to D are non sequiturs. Transition D to E is missing the one key ingredient. You can add it easily, with at least a hypothetical form of Alzheimer's, though it's still not exactly what you need to do.
Let's back up. We're not merely talking about "continuity" in the sense of continuous transformations. What we are talking about is continuity of the self. Before we get into that, let's discuss what the self is.
We are separate agencies. I have thoughts, and you have thoughts. I see things, and you see things. But I do not think your thoughts, nor do you think my thoughts. I cannot see what you're currently looking at, and you cannot see what I'm currently looking at. We each have our own respective set of subjective states. (This also gets into intentions and so on, but I'll leave that topic for another day).
Now as well as having separate experiences, we remember separate experiences; furthermore, we remember our own experiences. Not only do I not think your thoughts, but I do not remember thoughts that you have (that you don't express); and likewise, you don't remember thoughts that I have (that I don't express). I cannot remember what you see and you cannot remember what I see. But I can remember some of my thoughts, and remember some of the things I see; and you can remember some of your thoughts and some of what you see. (Ignoring for now how flawed some of these memories can be...)
In other words, just as I am a separate agency from you, I remember being a separate agency from you. And I remember being a particular agency with particular experiences--particular thoughts, particular sights, and so on. In other words, in the past, there was a separate agency; in the present, I remember "being" that agency, in that I remember some of the things that this agency to the exclusion of all other agencies was privy to. And
that is the continuity of the self.
You can, hypothetically, remove those memories by changing my brain states. If you do, you remove the continuity of the self. Right there in scenario #1, before you get to your "problems" of #2 and #3.
But I might also add, that you're going to have a hell of a time making my brain state match yours by simply removing experiences. There's no way you can just make me forget this, forget that, forget that, and so on, and wind up making my brain state match yours. To do that requires some addition; Alzheimer's just won't do. You're going to need some neurosurgery for that one. So if you want to argue that you can make me go from state A immediately to state E and that therefore I am the same as you, you don't get to do that with a scenario #1.
You can, however, involve a neurosurgeon and perform this sort of transformation. If you like, you can perform it very gradually; take your 70 years, and slowly transform my brain state into you.
Once you do that, the entity you get winds up being a version of you. Though it's not going to see what you see, and so on, so it'll diverge quickly (the problem with universal symmetries--they're only applicable in general in principle; in particular, move me one foot to the left, and I see something different than you). Nevertheless, if you want to argue that because you can slowly transform me into you, then we're both the same, then you're simply committing the fallacy of the heap.
So there you have it. Continuity of the self involves, by my account, an actual memory tie to your "being" a former self--specifically, a former particular agency in the world. Continuous transformations from one person to another simply changes the self from one person to another, if you do it. (And if you want to build a chimera brain, you'll just get a chimera self).