If you go, they go
If something were to happen to the President while they were in office, for example, an assassination or a natural, but unexpected death, their family must leave the White House. You would think that they would let them stay for a little bit longer, but they need to leave, within a reasonable time frame of course.
When John F. Kennedy was shot, his wife and family had two weeks to leave the White House and make a home somewhere else. We guess this makes sense, you need to make room for the new leader to come in, after all, there is a nation to run and things to get done.