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The problem lies in the fact that in most airports, as you exit Customs/Immigration, you are also exiting security. The connection time is irrevelent.

And as such, you have access to items which would be permitted as a checked bag, but not in a carryon. Things such as firearms come to mind.

While the bag will be claimed at JFK for customs/immigration purposes, the bag should be tagged in EVN all the way through to MIA. I am not familiar with JFK, but there may be a baggage recheck area immediately outside of customs, where the bag can be simply dropped and the passenger would be on their way.

I hope I have made some sense here.