The Riding School runs Gymkhanas most bank holidays, which are a mixture of different events.
We have done Dressage competitions in the past as shown here:
http://www.chestnutsridingschool.co.uk/image-galleries/shows-and-events/dressage
Here are some photos from a Jumping Orientated Gymkhana:
http://www.chestnutsridingschool.co.uk/image-galleries/shows-and-events/bank-holiday
We always provide medals to everyone who takes part, not just the highest places.
 
As for outside events, people who have their own horses at the Livery Yard, sometimes do take them, but Riding School horses are not generally allowed to take part in outside activities.