It is genetics. You cannot train a horse to gait who does not already have the genetics to be ABLE to gait.
I am unfamiliar with the American gaited breeds; American Saddlebred, Tennessee Walking Horse, Racking Horse, Rocky Mountain Horse, Pacers et cetera...I have only recently educated myself on the Peruvian Pasos and Paso Finos, as those are the two gaited breeds I have owned in the past 4 years.
In 1493, Columbus' second voyage to the Caribbean, he brought horses from Spain to Santo Domingo. They were of Andalusian, Spanish Barb and Spanish Jennet descent. The Andalusian and Barb horses provided the phenotype (the look) and the Jennet provided the genetic propensity to gait smoothly, genotype (genetics). The Andys and Barbs are thriving today with a loyal following, but the Jennet is extinct.
Horses were taken to Puerto Rico by Martin de Salazar in 1509. Diego de Velasquez invaded Cuba with eight horses and mares in 1511. 1512 saw horses at the Isthmus and in 1517, Cortes took 17, including one foal born aboard ship, to Mexico.
The horses that Columbus and the other explorers brought are the foundation of our native USA Mustangs. Sometimes you will hear stories of the Pacing Stallions of our American West, these stories are about the gait that managed to carry through from those first horses the intrepid explorers brought with them.
Anyhow, for 500 years, the islands of Puerto Rico, Cuba, Dominican Republic and the mainlands of Colombia and Peru have been breeding these horses with little outside influence. This has managed to keep the gait genetically within the breed.
Our USA Mustangs have experienced many outcrossings of diverse domesticated horses, turned feral or purposely turned loose, of TB to draft blood and I feel the gait has been lost through those crossings.
In Peru, (Peruvian Paso) the breed is known for the wide swinging front legs, called "termino", a movement actually begun at the shoulder. This is a variation of the gait that was developed from environmental influence. In the Pampas (area where cattle are run, very swampy) the horses had to lift their front legs over the mud and learned to swing them out to the side before putting them down. The Peruvian Paso also developed a larger body mass, though the phenotype still looks similar, with rounded bodies, high head carriage and luxuriant manes/tails. This resulted from their geographical isolation and different grazing available for the land.
The Paso Fino travels without termino. I called it lateral in the above photo of the horse with 2up/2down legs because in that photo it
appears to be lateral (same side) movement...but the gait is actually "isochronal", which means evenly spaced. The Peruvian and Paso Finos have a 4-beat footfall with evenly spaced steps. It sounds like: taca taca taca. Because it is an evenly spaced 4-beat gait, it is not a pace, which
is truly lateral. The pacing horse lifts both legs on the same side at the same time and sets them down together at the same time, making a 2-beat gait. The trot is also a 2-beat gait, but the diagonal legs move together.
Back to 4-beat gait.
This gait, being isochronal, is actually BOTH lateral AND diagonal. There are periods in the gait when the horse is supported by two legs (sometimes the same side, sometimes diagonal sides) and times when it is supported by 3 legs. The legs appear to move in a wave...the R hind lifts, then the R front lifts. The R hind sets down, the L hind lifts up, the R front sets down and the L front lifts up and begins the process over again.
All horses can walk (
called a flat-walk by the gaited breeds), trot, canter and gallop. Gaited horses are able to do the 'extra' gaits, basically a 4-beat gait called different things by the different breeds; Icelandics call it "tolt or flying pace", Paso Finos call it "fino, corto and largo" (3 different speeds), Peruvian Pasos call it "paso llano and sobreandando" (2 different speeds), then it is also called "racking" or "running walk" in our American gaited breeds. The one thing these horses have in common, is their ancestors trace back to gaited horses or the Spanish Jennet. If the genetics aren't there, the horse cannot gait.
Was that too much info?