In OBR, I think that when the BR misses 1B and then overruns the bag, he has to be tagged whether the action becomes "relaxed" or not. At least that's the way I've seen it handled.
But I can certainly conjure up troublesome scenarios, e.g., BR misses 1B, overruns by 35 feet. Nobody appears to be appealing, and he asks the ump for time while he pulls up his socks. Grant time and you prevent the defense from appealing the miss; don't grant time and you have indicated that something's up.
In that other scenario about chasing the runner to 2B, yes the fielder could do that. However, I think an appeal could be made even during unrelaxed action, as long as the runner who committed the infraction has continued past the missed base, has left the vicinity of the base, and displays no intent to return.
Batter hits a ball off the fence, misses 1B, continues around 2B and 3B and is on his way home. The defense throws the ball to 1B and appeals the miss. That's not relaxed action, but I'd uphold that appeal.
I'm sure somebody on the board can give you a definitive answer.
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greymule
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