The order does not purport to authorize U.S. agents to enter Mexico, even if hot on the trail of terrorists who cross the border to avoid imminent capture.
Ultimately U.S. military members might be ordered into border and terror operations in U.S. areas, and efforts against Mexico may escalate, but for now the order is limited to what it says. Trump can ammend it, revise and amend its reach, or issue another Order. For now, it is alone as to "cartels" whatever they are. Expect that to be litigated so that a judicial definition will evolve, specific entities may be designated as "cartels" under the order, etc. It is a first draft effort to give notice of more to follow.
Will Hegseth send troops to the border, will some troop segment or group cross the border, and will that be diplomatically resolved? It is likely if the question arises Department of State will negotiate something with Mexican comparable authorities.
First step has been issuing the Order, giving notice even if vague, and no U.S. officials are likely to express much opinion in support of "cartels," who have been a near constant target of press disapprobation.
The order is there, what grows from it is to be seen, and it is expected that in the first hundred days something related to that order will arise.
UPDATE: Related: https://www.whitehouse.gov/presidential-actions/2025/01/guaranteeing-the-states-protection-against-invasion/
Are we being invaded? Does saying so make something a reality? Is there a ready answer or must we wait to see how things shake out?
The new administration has edited whitehouse.gov and that means what?