Well, they don't actually "stay" in the original district, as once you've bought and used them, they're out of the game.
Well, yes... that is what I meant in an opaque way.
I was trying to make it clear that when you buy dice, you don't *move* them to your district & therefore you cannot re-roll/flip them since the rules says you can only do so with dice in your own district. So technically, they *stay* in their district until they've been used up.
But, of course, they get removed from the board once used, whether they were yours originally or bought.