the IOC has limited equestrian sport in the Olympics to a maximum of 200 human ATHLETES, IIRC. It may be less. The FEI has to figure out how the IOC number of horses will be allotted to each sport. Then, since there are always more nations wanting to compete than there are horse slots available, the FEI sets qualifying criteria for individual entries and also for teams. It is possible that a country (as Australia did last cycle) can qualify enough individuals to create what is called a composite team.
So WEG was the qualifier for three dressage teams–the top three. Three more will be determined in next years European Championships from those that haven’t already qualified. One team will qualify from all the Americas. Brazil gets a team as host country. One team will qualify somehow from all of East Asia and “Oceania”. That’s nine teams of 4 or 36 horse and rider combinations, but for some reason I think there will be ten teams. Maybe there is one slot for Eurasia–Russia and the Middle East. Each team is, IIRC, allowed a reserve horse, so that’s another 10, bringing the total to 50. And there will be individual riders who have enough FEI points to qualify from nations who won’t field teams.
But all of that is because the IOC says how many horses in total will be allowed to go to Rio.
We can qualify a team in one of two ways–officially by winning at the Pan Am games, or unofficially as a composite if enough of our riders get enough points to qualify as individuals. But the composite team doesn’t have to be the individual riders that have amassed the points to qualify.