These articles give a little behind the scenes look at the different plane set ups for flying horses. In some of them, the horses do not look anywhere near as enclosed as the ones I’ve seen.
I have also traveled on planes where the horses were in the crates in the back of the plane, and the front of the plane was full of passengers who had no idea there were horses on board. So when I went back and checked the horses during the flight and came back to my seat covered in loose hay, the people sitting next to me gave me a little bit of a funny look. Lol.
https://www.usatoday.com/story/travel/flights/2016/10/26/tex-sutton-air-horse-one/92782198/
This is a video the FEI put out five years ago for WEG. It gives a pretty good look at the general process when the horses fly internationally.
It’s hard to imagine how a horse could have gotten out of one of those boxes inside the plane, but again… horses.