I don't do ringy type showing, but I do know how to cool a horse.
Get a spray bottle with some water and spray their necks, under their necks, and between their back legs. Not too much water; just enough to be damp to the touch. Either stand them in front of a fan or fan them yourself. Your goal is to keep enough air moving over the water so that the wet spots are cool to the touch. This is something you can do between classes when you're waiting in the heat and after classes when your horses are winded and hot.
Keep your girth loose when you're not about to ride. Undo the girth occasionally and lift the saddle a couple inches off their backs and let it back down. So this a couple times. Seems silly, but a little fresh air between the saddle and a horse's back seems to make them a lot more comfortable.
Of course, offer cool water frequently. Let them drink as much as they want, even if they're still breathing hard. Horses cannot colic off water, no matter what your grandma told you.
If your horses are waiting untacked in a stall, spray them with your spray bottle lightly all over. Keep a fan on them; dead, unmoving air is so much hotter than moving air.
Something I like to do during long waits is get a big old towel, soak in it in ice water, and tie it around the horse's necks. The neck is one of the best places to cool a horse, especially the bottom of the neck. It's kind of like when you put water on your wrists; a lot of blood runs through there and makes you feel cool all over.
PS: whenever you put water on your horse, make sure the water isn't hot. If a horse is hot enough itself, the water can go from cool to nasty hot in 30 seconds. Get a sweat scraper and remove all hot water. Add new cool water. Letting hot water sit on your horse does more harm then good. Water is an amazing insulator and will sit on your horse like a hot, muggy blanket.