when can a runner leave the base

Question
We have moved and my girls are on a fast pitch team. I can understand some rules are local rules but aren't most of them universal? For instance, when is a runner allowed to leave the base? I thought it was when the ball crosses home plate but I was told today that it is when the pitcher releases the ball.

Answer
For most organizations it is when the pitcher releases the ball.  Some local rec leagues, for the younger age groups, don't allow the runner to leave the base until it crosses home plate.  

Little League doesn't allow the runners to leave the base from 12U (Majors) down until the ball crosses the plate.  After this it is on the release of the pitcher.  

This rule change is done to even the playing field for younger catchers to give them a chance to throw the runner out at second on a steal.  Most young rec league catchers wouldn't have a chance if the runner could leave at the release of the pitch.

You have it right about most rules being universal.  It is just the opposite of what you thought on this one.