In Washington, Republicans called the president's deal, struck late Friday, a masterstroke of negotiations.
And for all Mr. Trump's tweets of triumph over the weekend, few immigration experts, citing the length and porousness of the border, believe it actually is possible to stop immigration from south of the Mexican border completely.
Trump tweeted on Sunday that the Times report was false, adding: "We have been trying to get some of these Border Actions for a long time, as have other administrations, but were not able to get them, or get them in full, until our signed agreement with Mexico".
The Trump administration had been careful not to set any concrete goals, so judging success will be tricky.
In the month of May, border officials reported the apprehension of over 132,000 illegal immigrants crossing from Mexico into the United States. Homeland Security officials predicted the overall numbers would remain above 100,000 for months into the future, so one definition of success could be cutting those numbers below that mark.
"This is likely to have only a small impact on solving the root causes of Central American migration because numerous components are things Mexico had already said they would do", Schumer said on Twitter.
In 2018, an average of about 50,000 people were snared at the border each month, meaning Mexico would have to achieve a reduction of almost two-thirds of the flow in order to meet that target.
The deal would also send the Mexican National Guard police force to its own southern border, where many Central Americans enter Mexico.
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A "US-Mexico Joint Declaration" released by the State Department says the US "will immediately expand the implementation" of a program that returns asylum-seekers who cross the southern border to Mexico while their claims are adjudicated.
The goal is to change the incentive structure.
The program was challenged in court earlier this year by the American Civil Liberties Union (ACLU) and other rights groups which say it puts asylum seekers in danger and violates USA and global law.
That program, announced in December, would be expanded across the entire U.S. -Mexico border under the terms of the agreement, according to the State Department.
Farm states, among the strongest of Trump's supporters, have been hit hard by the president's trade war against China, and the threat of additional action against Mexico had some farm-state senators up in arms. It will offer work permits, health care and humanitarian assistance to the migrants who are waiting on Mexican territory.
As of last week, Mexico had received almost 10,500 asylum seekers, who cross back into the United States when called to attend immigration hearings.
The decision - announced by tweet late Friday - ended a showdown that business leaders warned would have disastrous economic consequences for both the US and one of its largest trading partners, driving up consumer prices and driving a wedge between the two allies.
That said, the Wisconsin lawmaker added, "I think he used them as leverage in this situation brilliantly".
Democrats were conflicted in their reactions. Contrary to those arguments, there were no real new measures that Mexico has agreed to in this deal, instead, the best that can be said is that they agreed to get it done faster, how much faster is a question we don't have an answer to.