I don't understand this "rush to out" reasoning.
You always allow all playing action to complete before the next at bat starts. If the defense saw the missed base, they will be calling for the appeal ASAP. The umpire (IMO) should rule on this appeal before turning his attention to the next batter due up.
If the defense dilly-dallies with the appeal, and the umpire calls for the next batter (explicitly or implicitly), then he can rule the absent batter out.
Side question: can the appeal still be honored even after the absent batter has been declared out? I'd say so since no pitch was thrown.
As to the inning-ending out situation (e.g. BR enters the dugout before touching 1st). The rule book says the new half inning begins immediately upon the 3rd out of the previous inning. Once the 3rd out is recorded, the team is immediately on defense, so none of their batters are due up until their next offensive half. Therefore, the absent batter's out is by rule the first out of the team's next offensive half inning.
That's how I see it, anyway.
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Tom
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