To start, gait in the soft gaited breeds is bred in, not trained in. You can train a trotter to “rack” and you can train a pretty quick walking cadence but the true running walk or marcha or other soft gait is genetic. You can refine the gait with training or you can change it with leather and iron (tack or farrier alteration). That’s whole other subject, though.
Some very laterally gaited horses have great difficulty with the three beat canter and some don’t. All can learn it, however, and learning it will improve gait in any gaited horse, not detract from it.
I’ve observed in 30+ years with gaited horses (mostly TN Walkers or Mangalarga Marchadors) that it is not uncommon for the horse to gait better in contact than on a loose rein. I’m not sure of the biomechanical reason for this, but it may be that the more lateral the horse the more they will “balance” against the pressure of the bit.
Try this experiment. Get down on all fours (on a carpet if you can for comfort) and crawl around for a minute or two moving as any horse would move at the walk. Now move diagonal pairs, as a trotting horse might. It would probably be a bad idea to go for a moment of suspension! 
Note that you can easily balance on two, opposing “legs” and that the transition from left to right diagonals is pretty smooth. Now do the same thing with lateral pairs, moving left arm and leg at the same time. Note that to balance against gravity you have to “lock” your back and roll slightly to the right to keep your balance. As the left pair return to the ground you must again lock your back and roll left to permit the right pair to lift off and move forward. This generates a gentle “rolling” motion often found in laterally gaited horses.
As part of this balancing a horse (with a lot of body weight in the head and neck) will use that weight to smooth out the dynamic forces generated by his much larger weight and speed. A human can simulate the back, but without that head and neck can’t simulate much about the front. IMO the horse will use the bit as part of its balancing mechanism, if it’s available. This might explain a commonly noted phenomenon of horses trotting at liberty but gaiting under saddle.
On the other hand some horses gait all the time, whether or not at liberty or with tack.
We ride in a military style (a la the Cavalry School at Ft. Riley or Samur) as opposed to a “dressage” style commonly seen at show grounds. Our style is always to ride in contact but the level of contact is always adjusted to the circumstance in which the rider finds themselves. It’s also a mostly one handed style (one hand for the horse, the other for the lance, saber, or pistol). This means that reins are not regularly adjusted but hand position and rider balance is. The double bridle is used to teach the horse and rider these skills and is used in parades, demonstrations, etc. In the field (at least in the 20th Century) the curb was generally left in the barracks and the bradoon was used by itself. The base riding style puts great emphasis on use of seat, leg, and balance before using rein pressure. But the horse is taught to respond to either. The saddle is not heavy or complex and permits reasonable feeling of the horse’s movements (even if loaded for the field with weapons, cantle roll, pommel role, and saddlebags).
In this we are similar to the style of the working cowboy who has one hand for the horse and other for his rope or other tools. Depending on which era and geographic style you’re dealing with the cowboy might be using a snaffle but more commonly you might find a curb with some weight on the reins and the reins, themselves, slack. Seat and leg pressure are less emphasized due to the size and weight of the Western saddle that has to stand up to the pressure of roping a full grown steer. Some horses are very “cowy” and will cut or herd instinctively with the rider just telling them which animal, or group, to aim at and then let the horse do its job.
The show ring practices you sometimes see in gaited horse shows can be monstrous. Heavy hands and passive seat and legs are VERY common. Horses trained to this standard can get “dull” pretty fast and re-habbing them can be a process that requires time and finesse.
There are a wide variety of gaited horses in the U.S. and just what you have under you at any given time can vary greatly. A horse that is very lateral will require a different approach than one that is very diagonal (and while lateral in North America is the most common type of soft gait the farther south you go more diverse will be gait styles).
To the trotting trainer new to working with a gaited horse I’d say drop back to the most classical training you’ve ever received and start there. Let the horse do what it does until you get a feel for it, and then use the classical (not necessarily modern, competition Dressage) riding aids to begin to “connect” with the horse. If the horse is young and not well broke to the gaited techniques common to the community your job won’t be all that different than with a normal trotter, just much more comfortable! If the horse is older and more steeped in “hand and heel” riding then you will have to overlay some bad habits with better ones. It can be done but not likely done fast and will require several “wet blankets.”
If you want to produce a comfortable horse for trail or fun then you won’t have much problem. If you want to produce for blue ribbons in the gaited show ring you’ll need to work with a professional gaited horse trainer (and you might not like what you have to do). If you want to train for a dressage-type environment you should probably work with a gaited horse trainer that is interested in that type of work just to get a good start for yourself.
Good luck in your program.
G.
