Neither.
The pitcher has to return to the rubber in order to end a play, but the fact that he's returned doesn't end continuous action.
A common play in LL is R3, batter walks and runs through first heading to second trying to draw a throw so R3 can score. If the pitcher gets the ball and toes the rubber before the BR reaches first, the BR does NOT have to stop at first, but can still continue to second, provided they do so without stoping after reaching first.
If BR rounds first, goes partway to second, stops, and starts dancing around trying to draw a throw, when the pitcher toes the rubber, BR must make a decision where he's going. If he doesn't immediately do so, he must return to first.
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