They updated the rulebook within the last few years (idk when, exactly) to explicitly state that runnings were allowed. Previously it said “ Martingales of any type are prohibited in Under Saddle, hack and tie-breaking classes. Standing martingales are allowed for all over fence classes. All other martingales may be considered unconventional."
Now it says:
HU105 Tack
1. Snaffles, pelhams and full bridles, all with cavesson nose bands, are required.
a. Judges may penalize, but may not eliminate, a horse or pony that competes in a snaffle, pelham, or full
bridle that is unconventional. Unconventional snaffles, pelhams, or full bridles include, but are not limited
to, hunter gags, kimberwickes, etc.
b. Judges must eliminate a horse or pony that competes in illegal bits or nosebands. Illegal bits include, but
are not limited to, three rings, gags (other than the hunter gag), et cetera… Illegal nosebands include, but
are not limited to, drop, flash and figure eight nosebands.
2. Curbs: When using a curb with any leverage bit, it must be constructed of loose links, joints, and/or lie smooth
against the jaw of the horse and be free of twists, sharp objects or anything inhumane. A curb may be wrapped or inserted into a cover for the comfort of the horse. A curb may not be used in conjunction with wire, metal, rawhide, metal “keepers”, or any other substance except for attachment of curb to the bit.
3. Horses must return to the ring for conformation or soundness wearing a snaffle, pelham or full bridle with a cavesson noseband.
4. Martingales of any type are prohibited in under saddle, hack and tie-breaking classes. Standing and running martingales used in the conventional manner are allowed for all over fences classes. All other martingales are considered illegal. A judge must eliminate a horse or pony that competes in a martingale other than a standing or running martingale used in the conventional manner.
a. Cable or metal tie downs are prohibited for use on horses and ponies