Obviously you sign a contract which permits them to put the animal down as per veterinary advice if something should happen to it. What does my head in when people talk about shipping horses on airplanes is when someone talks about this as if it's a common occurrence and something you seriously need to be concerned about. It's not.
I've never shipped a horse from the UK to the US so I don't know what they test for other than Foot and Mouth, which is their main concern as it's something North America doesn't have and certainly doesn't want. There are far more diseases in the States than here. To fly here, we had to show proof the horse was vaccinated for a rather lengthy list of things (she would have been anyway as all those are routine vaccinations) and get special tests done for West Nile, vesicular stomatitis, and EIA. The horse did not have to be in quarantine before or after those tests so long as they were done within a specified period before she traveled (I can't remember what it was... sixty days? Ninety days? Something like that).