They may not progressively replace themselves, but some die and some new ones occur. Worse, on the cellular level the dendrites change with some frequency. Even worse, we don't understand exactly how the cells decide which dendrites to fire in response to stimuli (each cell can have thousands), and - here's another fun part - the brain doesn't stop. Unless the "transfer" is instantaneous, you're not going to get a good copy. Add to all of that the fact that hormones influence the functioning and you get a real mess.
The fact that it's a copy demonstrates that it's not *you*. If the process can be made noninvasive, you inarguably get a (somewhat) mental clone, not *you*.
If I chose to do this "transfer", you can bet it will be the very last friggin' thing I do in life before death at old age, not an option I'll consider a moment before. Kinda queers the whole "getting rid of the geezers" thing.
A simulation, regardless of how detailed, is still a simulation. Anything which destroys the human body during transport (including traveling great distance or time) is not a good idea if you value the integrity of self.