In Czech we also have draft horse competitions - I don't know how to translate the name of the first discipline "formanská jízda" - literally it's a "farmers ride". It's a competition resembling to a driving marathon - pair of draft horses pull a four-wheeled carriage as fast as they can and have to make it through some obstacles, made of poles, bridges, watter crossing and such.
To my knowledge, very few draft horses participate in these types of competitions. Many don't want their horses to do anything faster than a trot in the harness. At one of our local shows, we do have a farm cart class where we are required to show our horses at both walk, and trot in almost a mini-dressage test of sorts. Showing how well they control. As well as being able to both gee and haw. My one mare is great at this and I took her single in my farm/training cart. She sidepasses both directions exceptionally well. She's also very responsive. She'll canter in harness, but don't let my dad know that. He'd have a fit.
The second discipline is a log pull - one horse has to pull a long heavy log in a given track (with slalom and backing up). Third is a weight pulling - one horse or pair has to pull a sledge with increasing weight on it.
We do have horse pulling here in the US. It can usually be found at any draft horse related event, as well as county fair's. Mostly American Belgian's are used, however on occasion you'll see crossbreds or other draft breeds. The Belgians have maintained lines that are very heavy in body and mass, they market them exactly for horse pulling.
The last competition is a ploughing (with pair of horses; judge evaluates quality of the furrow, its depth, regularity etc.).
Do you have some draft horse competitions in US and abroad in general? In central Europe (Czech, Slovakia, Hungary...) it´s a common thing.
We do also have this, and I enjoy watching them. Sometimes we also have plow days, no judging, where people from all over will bring teams and work a field as a group effort.