Past the fact that it’s extraordinarily dangerous, most operating systems will not let you do this. Moving only moves files without changing them. This is the same problem with people wanting to continuously update one sound file every time they edit or make a change. Any error or machine problem at any time and there’s no show and no way to get one since there are no backups.
I would keep the old folder around and only delete it after you deliver the final work to the client. Even then, I have pre-production and raw capture work over a year old.