I suppose it really depends on figuring out what consciousness really is.

If we’re talking about pure digital information then nothing is actually moved, it’s just copied and written to a new location. From the perspective of the new mind, continuity would be preserved and it would think it had simply changed places. The old mind (if it survives the scanning/transfer process) would still be outside the simulation and see a new copy of itself in there.

On the other hand, if consciousness is some kind of understandable phenomena, with even more sci-fi tech we might be able to simply scoop it out and put it somewhere else. In that case it would be the same person.