Apparently there aren't age restrictions for ejections.
At least two bat boys have been thrown out of professional baseball games.
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June 24, 1948: Salisbury Cardinals manager Gene Corbett was booted from a Class D Eastern Shore League game and later used 13-year-old bat boy Paul Murrell as a messenger, instructing the kid to tell an umpire to "get in the game and keep your eyes open." Murrell relayed the message, got ejected too and sulked with Corbett in the clubhouse, according to the Eastern Shore Baseball Museum.