The pulling the elbow in that way can cause the Nursemaid elbow. But this in itself does not make it prone to the recurrent injury.
Please read this authentic resource on this about the causes and further details;
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The twisting of the arm if causes injury will result in pain and swelling immediately or after 12-24 hours. These minor trauma are not responsible for the Nursemaid's elbow. The Nursemaid's elbow is due to the ligament laxity in this age, predisposing for the slip and occur very commonly in this age group, children.
I hope this answers your query.
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