Actually, anything in both household and personal inventory will stay with them, even through saving them to the library. I've moved households that way often. It actually works better than an in-game move to another town, because when you use the in-game move to a new neighborhood, even with choosing to not sell their furnishings, some things always get lost in the move. My households lost their workout equipment every single time. Saving everything to household and personal inventory ensures that nothing gets lost.
