I've ridden mainly hunters my entire riding life and trained under several trainers, some better then others. (some significantly better) when I first started jumping I rode with a trainer for THREE years and never had any clue what even a leg yield was! We learned how to be pretty passengers and that was it. Thankfully i've ridden with some trainers who actually have a clue.... but yes, trainers like what Anabel described definitely exist, although not all are like that!
Basically after every diagonal line the horse knows it's going to change directions and changes its lead for you whether you ask for it or not. They generally don't just swap their lead whenever, it's mainly just in the corners when they know they're going to change directions. Some students just sit there and look pretty and don't ask, I think mainly because their trainers don't teach them anything else. I like to teach my kids how to balance their horses and ask for one properly, but have the 'back up' that in the show ring the horse will get it right if the kid doesn't. But I also like to buy made horses that just do their job for my beginner kids to teach them how to ride before getting anything remotely green.
I do find that most horses that don't get a clean change will change in the front before changing in the back, so yes, they will be on the cross canter. This is heavily penalized in the show ring, which is why auto changes are so dang important to people and why those horses are so much more expensive. Despite popular belief
hunter judges do not just look at a horse and say, ooh, that's pretty, I pick that one! Judging is numerical and there is a ranking of what is considered desirable. The best is for a horse to land on the correct lead after every jump. Next is to do a flying change. Next is a skip change (changes in the front, and then in the back, the longer the horse is on the cross canter, the more they are penalized). Next (and these are heavily penalized) is a balanced
counter canter. Next is to swap in the front and just cross canter around without ever getting the full change (or waiting several strides to get it). The worst is to break gait to a trot and do a simple change. Breaking gait automatically brings your score down from a 100 to 60, regardless of how perfect the rest of your round may have been.
I know dressage riders teach their horses how to change leads much later in their career so there's quite a bit more balancing/lateral work before it's crucial that they know it. They have the 'luxury' of not needing one until later. The second a hunter goes into the show ring (whether it's with a cross bars kid or the horse's baby green year) a lead change is necessary so people rush their horses into learning them.