Most people pack an extra suitcase just for the fursuit. Some people have fancy custom foam cutouts to hold them in place to make sure they aren't damaged during travel, but chucking it in there with some padding will usually do the job.

Back when I used to party I knew a guy that would take a school bus converted into a ball pit across the Canada-US border to Burning Man. Imagine that search. He said it usually took 24 hours to get across. I don't know why they let him through tbh, I'd just nope him right back to Canada.