Cops can only operate within their own borders (like a NY cop cannot enforce the law in Canada) unless they’re granted special permission from Ottawa to the White House on catching high profile fugitives, even then: their powers are limited in order not to violate their sovereignty. Would their badge hold any weight while they are overseas (who have their own judiciary)?
That’s only speaking on official terms: but, what happens if an American on vacation (like off duty sheriff) witnesses or encounters criminal activity while they’re abroad? Are they allowed to detain that person or do they have to eventually turn them over to local authorities? For instance, they could flex a LAPD badge in Japan, but is that valid or considered null?
It would generally be considered null. Most interfacing between police departments is through Interpol, the FBI, or the State Department. There is almost no reason for a local cop to have any standing in a foreign country.
That said, for some strange reason, the New York Police Department has an intelligence arm that works in foreign countries and is coordinated at the city level. Because of that, it is possible that New York City has independent relationships with foreign countries that doesn’t go through the US State Department. This came up as former mayor Eric Adams was caught getting bribes by Turkish officials for leniency in permitting for its NYC embassy. So, it is possible that a NYPD cop could have more pull than the average American local cop because of these relationships.