Here's a pretty good video of a batida (diagonal) gait of the MM horse.
Pretty nicely conformed horse.
Some noteworthy things are that the rider maintains a very classical seat. She is in contact but the contact is quite light. Looks like the bit is a D-ring snaffle (can't tell the mouthpiece). She uses her aids very effectively and gets impressive performance from the horse.
The batida gait is diagonal and not as smooth as the picada gait which is more like a stepping pace. This horse does not demonstrate as much three-legged support as I like to see, but it's still an impressive way of going.
If you want to judge smoothness in a horse get a woman with long, loose hair to ride it. There are two parts of a woman's anatomy that are wonderful at demonstrating smoothness. The first is the hair. If the hair is "quiet" the ride is smooth. If it's not, then it's not.
The other anatomical point(s)? I'll leave that to the reader's imagination!
She also does well "training" the horse to the surf. She does something that more riders ought to do when "desensitizing" a horse to a new experience. She puts herself between the "scary booger" (in this case, the surf) and leads the horse by gentle, but firm, encouragement. It works and the horse is soon at ease in the water.
Haras 8 Virtues is known for producing and training Marchadors to be more like "sport" horses. One of my fantasies is to go to Brazil and spend a week in one of their schools. Maybe someday!
P.S. If you watch the credits at the end you'll note they identify the amazona. That is the Portuguese (and in some Spanish dialects) the word for "horsewoman" or "equestrienne." It has nothing to do with river!