^ Because we are oh-so-very pretentious and breed-biased, blah blah blah. :roll:
No reason you can't train a draft-cross on the barrels. Heck, I used to take a purebred 18-hand Belgian gelding through the pattern--entered him in the pony classes at the local playdays. We made some wiiiiiiiiide turns, but had fun doing it. But now, don't expect him to every be particularly fast or competitive. How successful he is will depend on how much he takes after the Percheron side, but regardless, don't think he's going to go beyond the local shodeo level. But it sounds like you're just doing it for fun, anyway, which is what it should be about.
Get him really broke under saddle first, if you know how to do that. W/t/c on both leads, stop, back, flex. Get him responsive to leg pressure, and get him moving laterally with the bit ("pick up" to move over). Practice turns with a "pocket" on barrels set up in the arena or the pasture, both in the regular pattern and randomly. Start at a trot when you go to train the actual pattern. Keep a hold of your inside rein, bend him, push him out, and pull through to finish. Remember your pocket. Reach down your inside rein, drop your outside rein. Remember to keep the shoulder picked up, and work on leads. If he starts rushing, stop, back, and make him wait before allowing him to turn. The biggest thing is don't push too hard/fast, too soon! Like this video shows:
Take your time, start slow, build your way up, practice RIGHT, rather than FAST, and have fun!
This gal knows how it's done: