Diagram: Firing an Assault Weapon — whether stationary or moving, the firer can always shoot the number of times indicated on its profile, up to the weapon's maximum range.
Assault weapons are used to support units as they charge forward into the attack. They either fire so rapidly or are so indiscriminate that you don't have to do much more than point and shoot. For example, a flamer is a weapon which fires a fan-shaped gout of burning fuel so it's just as accurate whether you are moving and firing or not. These weapons shoot the number of times indicated - whether you move or not and regardless of range. This means they are very good for assaulting things ... hence the name!