I know we have went over things like this before, but I'll ask anyways. I had this come up a few days ago. So, R1, 2 out, 2 strikes. Batter swings at low pitch that is not caught by catcher. F2 picks up ball and is looking to tag B1, but B1 is already about 15 feet away walking towards the 3rd base dugout. So I look at R1 and he is crossing the mound on his way to the dugout. I know when to call the BR out, but how far do you let R1 go before you call him out for abandonment?

Only worry about abandonment by the batter and call him out at the appropriate time.

Don’t worry about abandonment by the runner. He has not done anything against the rules. But watch to see if he is tagged before you call out the batter. If so, he is the third out.

According to the rule book (OBR), call it when it's "obvious".

According to Evans, it's not obvious until he crosses the foul line or gets to the dugout (depending on the which runner and whic dugout), iirc.

I was assuming the Batter would reach the dugout before the Runner would reach the dugout (1st base side) or the foul line (3rd base side).

Maybe that is not a good assumption !