How does promotion work when someone has experience in a similar art?

So I'm curious how promotion works when someone has taken the same or a similar art elsewhere and goes to a new place. For example if someone has taken bjj and gets to a purple belt, then goes and takes bjj somewhere else. Would they be able to start at purple another place?

Given the same scenario what if that person went to judo? Since the techniques are so similar would that person start off as a white belt or what would happen?