People aren't information, so much as processes. You can record the information required to run a process, but unless you are actually running it, there's no person happening. If you copy someone's mental processes and then instantiate them in two different places, you now have two people. There's no supernatural connection between the two, but they do share the same cause and effect relationship as you-before-sleep has with you-after-sleep. Which brings up another interesting thought: Did you know that most of your brain turns off for around 8 hours a day, only to be restarted by your suprachiasmatic nuclei? If having your mentality process turned off and back on again later is the same thing as death, then you die every time you fall asleep. I don't feel that that is death in any meaningful sense, and see no reason to treat any other stop and restart event differently. Real, actual death is when something happens where your mind stops and then never starts back up again.
That is probably the best way to consider that. You can't simply do a drag&drop of consciousness, because that's like terminating a game on one computer, running it on another and wondering why you aren't dropped in the exact same place and state you were in the first computer.
You could copy the savegame, but if, for some reason, the original computer exhibits buggy behavior because some hardware, identical to specs, has an errant dust speck on it or something, then even still the copy is not the original. Hence the requirement for continuity of self.
Piecewise replacement would work, but it does depend on the size of the pieces. You have to be able to smoothly maintain the process, so you'd have to add a replacement while keeping the original piece, re-route the connection from the original's input to go to replacement's input without signal loss, then and only then, once the original is deprecated for computing, remove it; then repeat the process for the whole thing.
There is some leeway, of course, since a lot of the brain is not essential for personality maintenance. And the main issue is signal loss; a sleeping, cryofrozen patient would likely be relatively easier to work with as a substantial section of the replacement parts would be idling.
I'd disagree on the part about your brain being turned off during sleep - if that was the case, you'd have to plug yourself into a respirator every time you go to sleep, for once. In fact, I recall that total activity-wise, a sleeping brain exhibits more, not less activity than an awake one, but can't get a citation for that ATM.
And on that note, it's not even that it DOESN'T turn on again - it's that it CANNOT. There is a saying among medical staff that you're not dead until you're dead and warm - that's because full-on death occurs when you get irreversible damage - to extend my original metaphor, savegame corruption - as brain cells are destroyed and connections are lost.