I usually do a walk trot 10 min ride and give them the next day off.
Nice long walk just to stretch all the muscles
I would walk them to let them stretch and get out of their stall.
I walk with my horse and give her some rest, let her eat grass and take care of her, give a cooling gel or clay after the show.
In my opinion, you can either go a calm walk with distended reins just to create a link with your horse and please or do nothing at all. You can also come see him, groom him and let him graze, that's the best thing you can do to please your horse after a great effort.
You could ride him the day after (lots of club horses are ridden the day after) because the ache's day is two days after a show but i don't think it would be great for your relation and the link you want to create with your horse. Also, it depends of the level of the competition you did.
I hope i could help you :)
give them a day off, but it depends of the intensity of the competition, some horses can be worked the day after the show. But honestly, it just depends on the horse