Most typically, you see hunter under saddle classes. These are part of the hunter division, and are judged on suitability, style, and movement of the horse - stemming back to the first horse shows that judged fieldhunters based on if they would excel in the hunt field or out hilltopping. Walk/trot/canter both ways of the ring, not typical at all to see any extras. You see “pleasure” type classes more at non-rated horse shows and, IME, are more about manners and creating the picture of a “pleasurable” ride in addition to the factors above, plus sometimes pleasure classes include a hack (2 small fences) or different “tests” of the mount like backing, standing quietly, etc.
equitation under saddle is a whole different ballgame; judged on the rider’s hand, seat, guidance, and style. USEF tests can and usually are used (1-19, or fewer for less advanced riders) so you can expect a collected or extended gait, sometimes riding without stirrups, sometimes countercanter or various lateral movements.
Tack wise, hunter riders generally ride in close contact saddles due to the more flat seat and more appropriate flap. But, it depends on the rider and the horse. Look up videos of large hunter under saddle classes (Washington, Devon, PNHS, the National) or videos of flat phases of large medal finals (Maclay or USET) to see what these riders and horses look like. It is a great education!