We had also same kind of event last August at Orimattila, Finland, in connection with fitness orienteering event. 8 participants.
Photos.
They had basic map cases.
My dad loved to ride his mule on trails and dirt roads. I would assume a mule would still qualify though wouldn't look nearly as fancy.
NACMO:
http://www.nacmo.org/
Seems very active, and has a very nice-looking website.
"the thinking horse sport"
In the last eleven years I've held about six 'equestrian orienteering' events for the local hunt club. They participate to raise funds for the club paying $75 for the afternoon event and a barbecue. Usually about 20 people participate in teams of three or four.
The first few events were held on the Torbolton Forest map - mostly sandunes. Recent events have been held at Hidden Meadows Farm. That is how we found the area that has been used for the 2010 Canadian and 2011 Ontario Middle championships.
I have the greatest respect for the work the riders have to do to get their horses to and event, then cleaned and cooled down and back to the barn after the event. Much more involved than just remembering one's compass, whistle, control description holder and dibbler.