Yeah, it would definitely have to be the same duchy each time, with the duke auto-changing to be the current ruler.
Maybe have the Duchy of the capital city be connected to the ruler. In the case capital being moved have the Duke/Duchess of that region lose their title and the old ruler duchy become Dukeless (to avoid automatically giving a duke a region that's about to fall or something if you moved it in panic).
If the king is killed or otherwise removed then have the duchy revert to a lordless one until a new ruler is appointed.
