From your description I think you are describing the extractor. It does move - it must move to ride over the rim of the cartridge. If you pull it toward the outside of the slide it should spring back. Also, the hook that captures the rim should be sharply defined - no chips or broken hook.

When you say the "third" round, are talking about when shooting, or when hand cycling the slide? Honestly, if the extractor works as described above and the the hook looks good, I'd be looking for another cause.

Out
West