There are several different contexts of "moving on" (particularly for the Filipino culture).
You want a change to happen to where you currently are. You feel that literally moving will get you to that experience. These can be related to moving to a new house or work area or even a new relationship.
You have just had an extremely bad experience and you want to remove it from your head, your circulation and lifestyle. An example is a death of a relative, a loved one, etc. After mourning, of course, the next item is to move on.
You want to start fresh with every situation. You want to carry on with a new agreement / understanding. This is true for post arguments. You need to move on and bury the sources of the problem and never go back to revisit it again.
So, when you're called to move on.. you know what that means!