I know that as an admin I am able to do this, but it seems mods can't.

I've given one of my mods the ability to move files between categories.  However, they are only able to move 1 file at a time.  He does not have the ability to select multiple files from one category and move them to another. 

Is this possible?

If they can move them, then they can move in bulk from the category listing by selecting more than one of them (using the  checkboxes) and then selecting move. Ive shown this as an example below