I'm on the team that movement abilities should not be tied to a damage component.

Because by doing so, they stop from being emergency buttons to move faster somewhere and become requirements to add into the rotation, even when you don't need to move.

The only situation I can see this working well with a damage component is in the case of SAM's Hissatsu movement actons, because there's a better way to spend the resources for it if you don't need to move.