whose call by correct mechanics?

[Andy]

With a runner on first and the ball hit to the outfield, the PU has the call at third on the runner that started at first. The PU should pick up that runner at second base and watch for obstruction as she comes to third.

The BU has the Batter-Runner all the way to third if the lead runner tries to score.

One thing that I always cover in pre-game is the times when the PU covers third, I ask if (s)he wants to move for any additional play at home, or if (s)he wants the BU to move to home. In any event, good communication between partners is vital in these situations.