Yes, I had to try several 1x1 round plates until I found ones that would come off the game tiles before they came off the animal. It can help if you put a thin piece of tape across the 1x1 stud, so it fits more tightly into the model. Cylinders would work better, but the steed ends up looking like it is flying.
Quoting J Hinkel
Very nice steeds. I especially like the hero ones (bird was interesting one!). Do they come off the tiles easily or does that round 1x1 plate stick? The SW's Hoth game looks like theirs are on cylinders, like the bats in the cave map, so didn't know if that was a problem.