No matter how they paid for it, Lopez & Harper would never be given permission to stop what ever they're doing and fly to a different country (they never specify what country John and Bailey are in) and participate in an unofficial investigation that has nothing to do with any cases in Los Angeles.
Just because he's a patrol cop in LA this does not give John any special privileges in whatever country he happens to be in. He could be cited for impersonating a cop and tampering with evidence by local law enforcement. No matter how "practically non-existent" they may be.