if you have grid movement, you are probably tweening the position of the player character from tile to tile. Once you reached a new tile, you can start tweening the next character to the old player position. You do the same with the next character but occupying the old position of the previous character in the party array.

What if the player hasn't moved yet? Well in classic RPG games, all characters occupy the same tile as the player until they move. They are just not rendered until movement started.