The thing about "moving on" from a coach who has seen some success is it always involves a two step process.
STEP 1: Fire the coach. That's the easy part. Perhaps too easy, because then comes the much more difficult...
STEP 2: Find a new coach who is clearly better than the guy you just fired. Not so easy, this part.
If you're an athletic director, beware of Step 2. If you fumble the new hire after you fire a coach with a decent track record, the next person who gets fired just might be you.
Fans, on the other hand, can always scream for the coach to be fired without worrying about any resulting repercussions.