In the biblical narrative, it says that Bethlehem was full from all the people visiting where they were born. But if so many people were born there, why weren't there more places to stay? It seems like a lot of people must have moved out. If so, why?
There are comments addressing your question in this post from earlier today.
From a historical-critical perspective, Luke's story about the census is completely fictive. There was no Roman census that ever required people to travel to ancestral towns, or even their towns of birth. All the Romans cared about was counting heads where people actually lived so they could tax them.
From a literary perspective, Luke is not necessarily implying that the inn was too full, but that there was no private space within the building (be it a residence or a house of lodging) where a woman could give birth. People slept communally in the same room, so a woman in childbirth would be brought to the stable not for lack of space inside, but for privacy (my source on this is The Complete Gospels: Annotated Scholars Version edited by Funk and Miller)