Mary and Joseph were not refugees, they were traveling to be registered in a census
But they were seeking refuge, correct?
Joseph and Mary were not refugees. Joseph was a self-employed carpenter, and he took on a pregnant wife, so he must have had to financial means to start a family.
The Romans called for a census, but the insisted that everyone would return to their home to be counted.
The reason there was no room at the inn is because there were a lot of visitors taking up all the rooms.
Joseph and Mary were tourists, not refugees.
When the wise men dropped of their expensive gifts, that helped them to get to Egypt and remain there until Herod died.