First things first. What is the reason behind having this trainer, train your horse? How old is your horse?? Is your horse being green broke?? Are you just having the trainer freshen up the horses skills? Are you having the trainer finish your horse out??
If you are having this trainer green break your horse, then the fact that the trainer is a reiner should not matter. I all horses go through the same steps when they are first learning what the saddle is, and what to do when the rider gives diffent damands. Look at it this way, its like your horse is going to school. They first start out in elementary school, leaning the basic, (what the saddle is, how it feels, the mounting, walking and stopping) Then they go to Jr. high (walking, trotting, and cantering, turning and a little bit of leg commands) Then they are in High School, (where the sharpend up the skills, like: neck reining, lead changes, more demanding leg commands and so forth) Then comes college, this is where the would go into dessage, reining, jumping, cutting, roping, barrels/contesting .... you get the point. This is like there major. So if you are having a person that does not have experience in gaited horses, work on your horses major, that is not such a good idea. But if your just trying to get your horse through high school then you are fine. As for do a lot of trotting and cantering. If this is a trainer that has been training horses for many years and has gone to school, they will lay out a program to fit your horses needs. Your horse should get equal time walking, trotting, and cantering.
I hope I answered your question! Good luck!