Ruling
 

Runner on second, 2 outs ,ball hit to 3rd, throw to first on time but 1st baseman pulls foot off of bag, field umpire called runner out. Appeal to home umpire who overrules and call runner at first safe. How far can the runner on second advance ? Please give ruling. Is this a dead ball out?

First, you'll get a lot of grief for your terminology here. This is not an appeal and the home plate umpire does not overrule the base umpire (at least not if they are doing their job.) In any case, after the plate umpire conferes with the base umpire and the base umpire elects to change his call, then the reversal of that call has put the defense in jeopardy and the umpire is to use discretion to make the situation right. I can't imagine the runner ending up anywhere except 2nd or 3rd in this scenario based on whether she was looked back or advancing.

Since the runner is safe, I'm going to say it is not a dead ball out. Though that isn't exactly a term of art.
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Sumayah live

The PU cannot overrule the BU.

The runner should be placed where the umpires believe the runner would advanced had the call been safe and the defense continued playing.

Since the initial call was the 3rd out of the inning, it is basically a dead ball