That's what I was seeing too- that he's trotting on the back half and walking on the front (more or less) which is a fox trot. That's why I asked if anyone else could identify it as such, because I'm not real confident in gait identification although I'm trying to learn.
My mare get bouncy in the the butt when she fox trots too. If I'm riding in the wrong saddle it actually jars my back. (And I'm used to to trotting horses). But when a regular horse trots the whole horse goes up and down. When my mare does an animated fox trot, only the back end goes up and down, which doesn't agree with my back. What I have learned that helps smooth her out is to take it slow and not encourage her to go fast (unless we are cantering or something). But as far as her gait goes, slower is smoother.
The fox trot CAN be smooth, but isn't always in my albeit limited experience.