Appeals are made during continuous action (unrelaxed action) all the time. For example, a fly ball caught, runner doesn't tag up. The throw to the base to nail the runner before he gets back to the base is during continuous action.
There isn't anything that prevents an appeal being made during continuous action. The batter hits a triple and misses first base. There is nothing to prevent the offense from appealing first while the batter-runner is still running around the bases. Of course if the umpire rules that he didn't miss it, the offense is screwed.
The offense could appeal this runner for missing the base he is forced to while the runner is scrambling back to the base, although chances of this happening in this situation is remote.
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