Actually there is a new television series in the 2021 seasion called FBI: International about a fictional team of FBI agents assigned to cases in Euorpe.
The series follows the elite operatives in the FBI's International Fly Team which is headquartered in Budapest.1 They are charged with locating and neutralizing threats against American interests, wherever they may be.
And there is Criminal Minds: Beyond Borders.
Criminal Minds: Beyond Borders is an American police procedural television series created by Erica Messer that aired on CBS from March 16, 2016, to May 17, 2017. The show is produced by The Mark Gordon Company in association with CBS Television Studios and ABC Studios. The production is a spin-off from original series, Criminal Minds, airing on the same network, and is the third and final series in the Criminal Minds franchise. Criminal Minds: Beyond Borders follows an elite team of FBI agents of the fictional International Response Team (IRT) tasked with solving cases that involve American citizens on international soil.2
CBS aired a backdoor pilot on an episode of Criminal Minds on April 8, 2015, introducing the characters with a crossover episode eponymously titled "Beyond Borders".1 The series was originally intended to premiere on March 2, 2016, but was pushed back by two weeks and instead premiered on March 16, 2016, and filled the Wednesday 10:00pm time slot, airing immediately after the original Criminal Minds
So fictional teams of FB agents do operate in foreign countries. What about real FBI agents?
Actually I found an official FBI site about their international operations.
Today, we have 63 legal attaché offices—commonly known as legats—and more than two dozen smaller sub-offices in key cities around the globe, providing coverage for more than 180 countries, territories, and islands. Each office is established through mutual agreement with the host country and is situated in the U.S. embassy or consulate in that nation.
Our legal attaché program is managed by the International Operations Division at FBI Headquarters in Washington, D.C. This office keeps in close contact with other federal agencies, Interpol, foreign police and security officers in Washington, and national and international law enforcement associations. International liaison and information sharing are conducted in accordance with executive orders, laws, treaties, Attorney General Guidelines, FBI policies, and interagency agreements.
And it goes into considerably more detail about how the FBI functions in other countries. And I guess that some fictional stories about FBI agents in foreign countries are more realistic than others.
Anyway, it would make no sense for an intelligence officer instead of a police officer of some type to be involved with hunting an internaitonal jewel thief in Red Notice. The only questions would be whether there is another police agency more likely than the FBI to get involved in this particular case, and how accurately are John Hartley's activities portrayed in the movie.