Personally the running animation looks fine. Its just like how real deer bound around.
i like the pose of the reindeer but agree maybe the animation could be improved eg. make the right leg go back less?
This is really cute. 
Run is a bit odd, seems like a few frames are missing. Very nice otherwise. 
Merry Christmas!
The BG is an NPA gradient, please correct that.
Merry Xmas to you too ;)
The background is definitely iffy...also, why is this submitted as a challenge again?
nice!