Baseball, by the Rules
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Caught Stealing
When a baserunner is caught stealing in the middle of a plate appearance, the first consideration is whether or not the caught stealing was the third out of the inning. If it was, the inning ends, regardless of whatever happened on the pitch that the runner ran on. The batter who was up to bat during the stolen base attempt will lead off the next time their team comes up to bat, but will start with a fresh count, as the plate appearance that they were having when the inning ended no longer counts.
What happens next can vary greatly: if the caught stealing was the third out, then the inning is over; otherwise the inning and/or plate appearance will continue, with or without base runners, as appropriate.