Another coach asked me about this situation that came up in his game today, and I wasn't sure on the ruling:

Runners on 1st and 3rd, two outs, a full count on the batter.
Batter swings and misses at a low pitch, but the catcher doesn't catch it cleanly. The catcher, instead of going to 1st, throws behind the runner to 3rd and the ball ends up in left field. Runner scores.
But the batter; thinking he is out, is in the dugout so the umpire calls him out BUT counts the run.

Should the run have scored?

The run should NOT have counted since the 3rd out was made on the BR before he reaches first base. No runs can score when this happens.

I agree with large, no run. The third out was made before BR reached first base. This is not a timing play. 4.09(a).