Yes. Free actions can be performed at any time (subject to DM discretion). It should be noted, however, that if you fail the check you don't finish the dismount until the mount finishes its actions for the round as move actions cannot interrupt another creatures initiative (not even an ally or mount acting on yours).
(I'm guessing you're going for the "leaping off the charging horse and roll to avoid damage" trick.) If a GM won't let you do it with a ride check, you could probably just fall off your mount as a free action and make a soft fall check to avoid damage/landing prone. However, if you have a military saddle, you might need to take a move action to unstrap yourself from it first.
Ride isn't really adequate to describe some of the stuff I saw the rodeo trick riders pull off when I was growing up. A common trick was jumping from the horse the long side of the arena, then running across the center, catching up with the horse as came out of the far turn and remounting on the other side. (I think you lost points if your horse had to slow down, but I was never completely clear on the scoring.)
My hands down favorite trick was dismounting on the left side, bouncing off the ground, vaulting over the horse and bouncing off the ground on the right side, all in a single motion without touching the horse in the middle. I saw one guy do it four times in a row.
Assuming he didn't have the magic saddle that lets you automatically succeed at fast mount and dismount checks, I'm guessing he had a Ride bonus of at least 24 to be able to pull that off.