Overthrow on tag up
 

0 out, runner on 1 and 3.

Fly ball catch, upon returning the 3rd, the centerfielder throws the ball into the dugout BEFORE the runner returned to the bag.

I know the runner gets one base for the overthrow... but is it the same in this case since he was returning to 3rd for his tag up?

What would happen if the runner crossed home plate and went to the dugout and the throw went into the dugout? does that run then count because of overthrow and deadball?

What you know just ain't so. Runners get two bases when a throw (other than from a pitcher in contact (or considered in contact) with the rubber) goes out of play.

And, it's two bases from the base last legally touched -- so it doesn't matter whether the runner is advancing or returning.

So, yes, let the runner re-touch the base. Under OBR, he can retouch, even if he's past the plate when the ball goes out of play, but once the runner enters the dugout, he's prohibited from doing so. If the runner touches the plate *after* the ball goes out of play, he's prohibited from legally returning. Under FED, the runner is prohibited from legally returning if he's beyond home at any time once the ball goes out of play.

If the runner fails to legally retouch third, the run still counts, but can be cancelled and the runner declared out if the defense makes a legal appeal.