Some of the discussion of the memory theory of personal identity--notably Williams's famous paper--focuses on the transfer of memories between bodies, and holds that according to the memory theory the person comes along with the memories.
I too used to use such thought experiments to argue against the memory theory. But now I wonder if doing so isn't unfair. The memory theorist must put an "in the right way" condition on the transmission of the apparent memories (quasi-memories in Shoemaker's terminology) that secure personal identity. Not every apparent memory as of things that happened to Napoleon secures identity with Napoleon, even if the apparent memories come from Napoleon's memories, or else the historian who goes mad and thinks he's Napoleon and has memories causally derived from Napoleon's via causal intermediaries such as letter by Napoleon really is Napoleon--or at least has ceased to exist via fusion.
Now, it is plausible that cerebrum transplants carry memories with them in the right way, since the memories continue to be housed neurally as usual. But it is far from clear that, say, scanning one brain and implanting the memories in another brain counts as a transmission "in the right way". In fact, it seems pretty plausible that this is an aberrant causal chain, just as much as in the case of the Napoleonically obsessed historian. After all, is there a significant difference in aberrance between the content of the memories traveling from brain to paper (Napoleon writing them down) and then to another brain and the content of the memories traveling from brain to a scanner to another brain that makes the latter less aberrant?
Suppose that such processes are indeed aberrant. And so the memory theory starts to look more like a brain theory of personal identity, except with the added proviso that only the memory-carrying parts of the brain count, and they count only as memory-carrying.
On the other hand, if computers can be conscious, then such processes perhaps cannot be aberrant, since they can look just like intra-computer transmissions. This suggests the interesting idea that the memory theorist of personal identity must either drop the "in the right way" condition on memory transmission, and accept unhappy consequences, or must hold that computers can't be persons.