In Catholic School I was taught that Jews viewed their place of origin as their home and not where they lived. So if your family traced its lineage to Bethlehem, you had to go to Bethlehem to be counted for the census.
I’m wondering whether this was true, why this belief arose and how it worked out. How did the Romans react to it? Did it create a lot of turmoil in Judaea during the census with everyone moving around? What did Jews do if they lived far away like in Rome?
This answer from u/kiwihellenist doesn't address all the specifics of your question but I'm going to guess you're thinking of one particularly famous 'traveling for a census story' which this does talk about.
If you've ever wondered how making people travel to a place they don't live for a census makes sense(us), well it doesn't! Something I didn't realize is that it's only in Luke, the other gospels have his family leave Bethlehem later (Matthew) or just mentions the debate (John).