This is should be a result of rotational momentum and is true moreso for pitchers with lower arm slots.
I really think that the fall off to the glove side that others have commented on is due to his starting position and stride direction. In the video, he seems to be using the black line on the floor as a reference. His target is aligned with the black line and he sets his pivot foot on the line as well. So, he effectively starts in the middle of the rubber and then strides towards the throwing arm side of the line. But in order to get his shoulders squared up to the target, he leans to the glove side. This causes his center of mass to shift to the glove side resulting in the fall-off to the glove side.
The first thing I'd do is have him move to the glove side of the rubber and see what effect that has on his posture. It should reduce the glove-side tilt.