By my understanding, the FEI provides the experts and specialists who then work with/in/alongside both the International Olympic Committee (IOC) and the Olympic Delivery Authority of the nation selected to run the Olympic competition. The IOC has very little to no equestrian knowledge, and similarly for all the other participating sports, so it relies on the various international sport governing bodies to arrange the actual events during the Olympic competition. The IOC accredited equestrian officials are usually rebadged FEI.
To volunteer at London 2012, we Eventing people were given a secret code to insert into our application form. This ensured only those with the correct experience were offered specialist roles across “the field of play”. Greenwich Park felt very similar to a normal British Eventing competition, with a few extras.