Can checked swing be a 'fourth out' appeal?

[JJ]

R1, R2, two out, two strikes on the batter. Runners stealing on the pitch, which is a check swing in the dirt. PU calls "ball", catcher throws to third to try to retire the stealing R2. Third baseman misses the tag, but he is touching the bag when he takes the throw. Defense appeals the check swing, and the BU says "Yes, he went".
Where is the out recorded? And who leads off next inning? As soon as it became a dropped third strike, it became a force at third base, and the third baseman WAS touching the base when he took the throw....hmmm....

I love this game -
JJ