It depends on the country sometimes but as LGCT is a known organisation countries which held the event always ready for it. I have been in 2 countries not much so I cant say detailed but as far as I saw, horses travelling with plane most of the time or if you are using a truck, you have te blood test at your stable and the quarantine is at your own stable. For example before we send our horses to Qatar, we are sending the blood samples, building a quarantine area with portable stables and when the time comes sending the horses. When they are coming back, they are waiting at Saudi Arabia as they go through SA to Qatar.
But for Miami let says, you are using the plane companies and Experienced grooms accompanied To the horses to look after their every need throughout the flight, and were joined by vets.
As per humans, the horses travel with their own passports which include all the key details and information allowing a smooth and organised process.
Once the flight landed horses transferred to a special waiting transporation trucks. Then the horses were settled comfortably into their purpose built temporary stables in the competition area where they were monitored and kept under US quarantine conditions.
I guess it should be similar for the other countries for big events.