How do X-shaped control surfaces work?
First, the precise terms is not fins, but control surfaces. Fins would be fixed but the four rear surfaces can rotate around a spanwise axis. By rotating, they can be positioned at an angle to the airflow, thus deflecting it and creating an opposing force.

Compare that to a V-tail on an aircraft. They work in the same way by creating a force normal to their plane. While this force has a vertical and horizontal component, the horizontal components of the single surfaces cancel each other when all surfaces are rotated by the same angle. Their vertical components will add up, creating a pitching moment.

When the left and right surfaces are rotated by the opposite angle, their vertical force component will cancel while the horizontal component will add up, thus creating a yawing moment which swings the tail around.


  If it were to pitch downwards, which fins would need to move and in
  which direction?


All four surfaces will move trailing edge down, so the resulting force will point up and pitch the whole craft nose down.