Assigning a household to a unit and moving them in are two very different things in the Doorways app.
Assigning a household to a unit leaves the family on the waiting lists, but assigns them a certain unit that they will move in to when it/or they are ready.
Moving-In a household signifies that the household will be removed from waiting lists and the unit is ready and the family will be moving/has already moved into the unit.
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