Ball 4, You're Out?

[BayStateRef]

Babe Ruth game (14-year-olds)

Tie score. Bottom 7th. Two out. Runner on third. 3-0 count. Next pitch bounces in dirt (Ball 4) and gets away from catcher. R3 scores. Batter/runner heads for first as teammates come out from first base dugout to celebrate win. Batter/runner never reaches first base, instead joining his teammates who are on the grass between home and first. Players line up to "congratulate" other team.

What is the proper ruling? (And please cite the rule to support this.) If batter/runner is out, at what point does the umpire call it? Must the defense tag the runner? If not, at what point does the umpire rule the batter/runner out?