Factual error: A dangerous prisoner like Esperanza would not be flown into a civilian airport. He would be transported to the nearest USAF base, in this case Andrews Air Force base, always assuming that he is being extradited to Washington D.C. As well as this his wrists would be shackled to a chain around his waist which would also be c attached to ankle shackles. He would also be escorted by half a dozen armed police officers.
Continuity mistake: When McClane grabs Marvin the janitor and pushes him against the lockers, McClane's thumb changes position on the gun, some shots it's on the hammer, other shots it's around the handle.
Revealing mistake: McClane jumps out of the helicopter at the end and lands on the wing of the larger airplane. The stunt man who does this then proceeds to move up to a long wire on the side of the wing. Any normal airplane would not have a long wire on the wing.
Audio problem: John is on the wing of the plane, questioning why he is there. His mouth isn't moving with what he is saying.
Factual error: The exchanges between the bad guy pretending to be an air traffic controller and the pilot of the doomed aircraft are not even close to realistic. Air traffic control communication is almost its own language. The FAA handbook says, in part: "Brevity is important, and contacts should be kept as brief as possible. Jargon, chatter, and 'CB' slang have no place in ATC communications."
Factual error: Transport aircraft do not have ejection seats; only fighters, bombers, and attack aircraft have them.
Continuity mistake: When John and his wife hug, his hand suddenly appears on her head.
Continuity mistake: When the general exits the plane with Colonel Stuart, nobody shuts the door. It is closed in the very next shot.
Continuity mistake: When Barnes and the SWAT team reach the Annex Skywalk, they get on the moving sidewalk. Before the terrorist stops the moving sidewalk by pressing the button, the sidewalk is already stopped, even though the light was still green at the time.
Answer: In the first movie he's a New York cop visiting his wife. In the second Die Hard, he tells the airport officer that he's LAPD and moved there because of his wife's job. In the third Die Hard film, he most likely went back to New York because of marital problems and became a New York cop again.