I don't quite understand what you mean by persisting before moving on, but I'll try to answer. The ghost is bound to their grave and, barring any awful bugs, they're always there and prime for haunting.
You can add it anywhere and the ghost will still haunt. This can be the cemetery, your backyard, an end table inside, a random community lot, someone else's house, or whatever else you can think of. You can bypass the hauntings by putting the graves in the mausoleum or your inventory, but they can still haunt if you decide to place the grave later.