I have found the ideas for conditioning a barrel horse is so varied.
I have had heard of people only long trotting because of the extention of muscles and building stamina but others think it is bad because of the "flick" at the end of extention causing lameness if done extensively. Some people lope circles while others think it's hard on the hocks and lope the arena or straight lines. Some people think that horses should exercised in the manner that they are used(like a race horse) while others disagree.
Personally, again I say, that you need to exercise both the fast twitch and slow twitch fibers(the air and stamina comes with that), on how you do it will be dependent on your horse and situation.