I have used a CaF2 crystal a few years ago in an experiment during my diploma work. I cannot give a definite answer, but I think the main reason is that you will basically burn a whole in the crystal over time if you use it always at the same spot. Furthermore, it will probably heat up and change its nonlinear properties over time. So you just move it around to avoid thermal damage and keep your white light spectrum more or less constant. We had a translational movement. It allows you to move the laser basically everywhere on the crystal. A rotational stage will always put it on the same locations on a circle.

Thank you for your answer, Kai. Did you guys put the crystal in a vacuum chamber? Is it necessary to do so? At what speed did you move the stage? Does the intensity profile change with translation? How did you prevent it?

Like, I said I worked at this experiment something like 7-8 years ago. It was certainly not in vacuum but I cannot tell you the speed. For details, you might contact Thomas Lenzer from University of Siegen. He is still running the experiment.