Recently, there have been two training horses that have an issue where they cross canter. In other words they canter on the correct lead but swap behind so they are cantering on the inside lead in front and the outside lead behind.
Two things happen when the horse cross canters. The haunches are lost to the outside and the shoulder collapses to the inside. Some horses do this because they lack strength and flexibility, so if the rider can help them canter in a united fashion, the horse can be corrected.
Correcting the horse is not about obedience in this case but about developing the strength and flexibility required to hold the canter lead. If the horse starts the canter as a cross canter - it is likely that he collapses his shoulder during the canter depart. In this case it is best to use some inside knee to push the shoulders out and a little more upright during the canter depart.
If the horse starts out in a united inside lead but then later looses his back end, the haunches are sliding too far to the outside of the canter circle. The best solution to this is for the rider to sit heavy on the outside on seat. This creates a bit of a block to the horse swinging his haunches out and helps the horse stay aligned so he can keep the inside canter lead in the back.
This process will help rather quickly and the horse will gain strength and flexibility. As the horse becomes stronger, the rider will not have to weight the outside of the circle and the horse will be able to maintain a united canter lead.
Good Luck with your horses and be sure to have a great time!