I once messed up like that - forgot to latch the back door with a horse in the trailer, but it turned out OK. We weren't going far, and the divider/panel was closed with the horse in the front of the trailer, so he couldn't go out. However, we've been a lot more careful since then.
I also know a fellow who was hauling a load of springers to a sale, and he'd gone maybe a half-mile down the road when one of the floorboards in the trailer gave way, and one of the heifers' hind feet fell through. Someone driving past noticed it and stopped him right away and they were able to save the heifer. She's probably 6 or 7 years old now, slightly lame, but not too bad. He hadn't been going very fast and she hadn't been like that long, otherwise it would have turned out differently.
One more story. My riding instructor once had a promising young horse that she loaded in her two-horse trailer (straight load, small trailer), and tied him up front. She was going down the highway when the back door flew open, the horse's hind feet fell out, but because he was tied he couldn't fall all the way out...makes my stomach turn over thinking about it. Of course she realized right away what happened, but they ended up having to put the horse down because he was so badly injured. The problem was that the horse was leaning on the door the main latch gave out, and she didn't have a second safety latch.
After we heard about that we had a second bolt-and-pin safety latch welded onto our trailer door.