Just having a look at Permutation City, it's a pretty close idea.

It uses the assumption that human consciousness is Turing computable: that consciousness can be produced by a computer program.

Look at Searle's Chinese RoomSEPWiki for few discussions about computers running human consciousnesses. The summary is that if you imagine a room with a man sitting inside and you want to know if he can speak Chinese. You can only communicate with him by writing down words on a piece of paper and him sending back replies. So you write down some Chinese and put it through. Very quickly you get a reply that sounds reasonable. Maybe you asked for their name and they replied with their name. You ask a few more questions and he replies reasonably every time. So you assume that he can understand Chinese. The man actually has a book with a list of replies. He reads the note, looks it up in the book and replies accordingly.

Okay, you say. What if you asked him the same question twice, because a machine would give the same reply to the same stimulus. Sure, the book can be expanded. Possibly in this version it's a computer program that reacts psuedo-randomly to stimuli, or even worse can remember what it's already replied.

There's a real problem to do with whether a computer can think, or if it's just reacting correctly to stimuli. What's scary is that one of the solutions to the problem is that humans can't think either; they're just responding to stimuli. So how would we ever know if we'd made a conscious computer system?

Anyway, the theory is known under a few names. Procedural Identity/ Diachronic Identity, where Diachronic is Dia (Through) + Chronos (Time).

I'd start by looking at Personal Identity as a wholeSEP, and then looking into Temporal PartsSEP. Temporal Parts is the idea that the parts of our body persist through time and make up us, but they aren't us.

You're performing an amazing trick right now: you're in two places at once. How do you manage to be down there, near the floor, and yet also be a metre or two up in the air? Well, it's not so very amazing: your feet are down there on the floor, and your head is up in the air. Having spatial parts enables you to be in several different places, and to have different properties in different places: you're cold down there on the tiled floor, and also warm up there by the heater, because your feet are cold and your head is warm. Moreover, having parts could let you be in the same place as someone else: if you shared a hand with a conjoined (‘Siamese’) twin, then you could both wear the same glove without jostling for space.

A survey of metaphysics / E. J. Lowe. is an amazing primer on metaphysics and what it is to have temporal parts and exist as a process rather than a substance. The key part of temporal parts is expressed here if you've got JSTOR access. It's called the paradox of 1,001 cats and it discusses how we are made up of lots of different parts but obviously none of them are sufficient for our identity. Metaphysics : a contemporary introduction / Michael Loux. I also found to be quite helpful in getting into it without being too confused.

There's also just diving straight into Identity SEP if you're feeling brave.

Also if you have access to this one, it's Philosophers of Process, who are some of the people who argue who Process Philosophy which leads into Procedural Identity. Also available here although I've only read some of it, so I can't vouch for it.

What got me into the whole topic was Heraclitus' claim that you couldn't step into the same river twice, it's referenced at the beginning of that article. The whole idea is that if you step into a river at one time, wait five minutes and step in again, you've touching entirely new particles. So how could that ever be the same river? If you're arguing for substance philosophy it really doesn't make sense, because the way we teach people to understand concepts like this is by pointing at the river and saying "River." then a few seconds later pointing to another bit and saying "River." until we understanding that despite the fact all the bits of it are different, they're all just a river.

With process philosophy you say "Well that's one process, it's a process of moving water". In the same way electricity is the continued process of electrons and you kinda get out of that problem quite easily.

Oh and the Ship of Theseus in general discusses the problem with discussing substance compared to process but that's relatively simple if you read anything on temporal parts.