On the face of it, the last one does not sound so bad, as long as the NYPD is operating with permission from the jurisdictions where they are operating.
Just like how a company wholly owned by New York City could have contracts with New York City, London and Paris at the same time.
Except cops aren't your Average Joe, they need to obey the local laws regarding a police force operations and without a lot of laws (or a suitable hand-waving) can't do anything more than an American tourist in the destination country.
Just like how a company wholly owned by New York City could have contracts with New York City, London and Paris at the same time.