Found my 2004 NSA rule book!
Apparently, batters may advance on a dropped third strike, just like any other age group (I think it was ASA that used to prevent this in their 10U rules).
The 10U baserunning rules relate mainly to stealing.
- Runners starting on first or second base may steal only one base per pitch, even if the ball is overthrown in an attempt to put out the stealing runner. However, if the throw
goes out of play the normal two-base award would be given.
- Runners starting on third may not steal home.
- A runner attempting to advance beyond the one allowable base is subject to being tagged out. If the runner safely reaches the next base, at the end of the play the runner is returned back to the correct base without penalty.
- A batter receiving a base on balls may attempt to continue and steal second base.
- Runners may only score on: A batted ball; A walk or hit batter with the bases loaded; Awarded bases when the ball goes out of play; An illegal pitch.
Hope that helps!