The batter/runner cant walk over in the grass by the dug out and be ok to run automatically. If a tag is attempted and the batter is outside 3 feet on either side of the basebath. I hope you are reading rules somewhere and not a liscensed umpire(not trying to flame

)...also the batter is out once he goes out of play which may or may not necessairly be the dug out steps, im not sure on the specific ground rules for each stadium in the league. Maybe im wrong but I know at the HS and College level the rule book dosent say "dugout steps". Theortically a batter/runner could have a dropped 3d strike walk to the dugout steps go down to the last step and then head to first if the steps are in play, and i've seen cacthes made on dugout steps before, and cacthes made with less than two outs that results in the fielder on the steps. According to the dugout steps rule than a runner on bnase would advance 2 bases because the ball was secured in play and then carried out of play.
Reguardless he should of tagged the runner, you learn that shit in coach pitch.