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Buckaroo Bugs?
It may be... do you know if there are any other shorts with the same theme/rider character?
Yeah, it's "Buckaroo Bugs". I think that's the only one that includes Red Hot Ryder, though. Any other western is likely to be Yosemite Sam.
^ There is also Nasty Canasta but the cartoon he went up against Bugs ("Barbary Coast Bunny") didn't really fit the OP's description
Edited by jormis29Thanks a lot guys!
I'm trying to find a Bugs Bunny cartoon in which he (Bugs) was chased around by a cowboy in a horse I think. I remember the setting was desertic, the cowboy was red-haired (I think) and was NOT Yosemite Sam, and there was a lot of comedic twisting and acting from the horse as well... I really donĀ“t remember much else, just that it was very funny when I first saw it. Thanks!