Q: Does the Cute Fuzzy Alien move after a Brain Transference? If so, where?

A: Yes. The Cute Fuzzy Alien (CFA) moves after a Brain Transference.

But where does it move to? If the person (let’s call them Player A) who had the CFA played Brain Transference, and moved to their left (to the position which would normally come after their previous position) does the CFA go to Player A in their new spot, even though their turn would be skipped, or to the next player who has a turn, which would be the person after Player A’s new location?

The CFA would move to the next spot at the table, even though, in this case, that’s still Player A, and it’s not their turn because they just took a turn that ended with them moving to that spot. The CFA would just stay with player A until their turn came around again, and then start moving when that turn ended.