Open this photo in gallery Arizona's Republican Gov. Doug Ducey, front right, meets with Arizona National Guard soldiers prior to their deployment to the Mexico border at the Papago Park Military Reservation on April 9, 2018. Ross D. Franklin/The Associated Press

Some U.S. National Guard members have started arriving at the United States-Mexico border with more expected as federal government officials continue to discuss what they’ll do about illegal immigration.

The Republican governors of Texas, Arizona and New Mexico on Monday committed 1,600 National Guard members to the border, giving U.S. President Donald Trump many of the troops he requested to fight what he’s called a crisis of migrant crossings and crime.

The only holdout border state was California, led by Democratic Governor Jerry Brown, who has not announced whether troops from his state’s National Guard will participate and has repeatedly fought with Mr. Trump over immigration policy.

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Under the federal law Mr. Trump invoked in his proclamation calling for National Guard troops, governors who send troops retain command and control over their state’s National Guard members and the U.S. government picks up the cost.

Mr. Brown’s spokesman, Evan Westrup, said California officials are still reviewing Mr. Trump’s troop request.

Mr. Trump said last week he wants to send 2,000 to 4,000 National Guard members to the border, issuing a proclamation citing “the lawlessness that continues at our southern border.”

Trump administration officials have said that rising numbers of people being caught at the southern border, while in line with seasonal trends in recent years, require an immediate response.

Apprehensions are still well below their historical trends during the terms of former presidents George W. Bush and Barack Obama, both of whom also deployed the National Guard to the border.

In Mexico City, a caravan of Central American migrants that had been heading north stopped in the Mexican capital. The caravan had sparked furious criticism from Mr. Trump, followed days later by his National Guard border-protection deployment plan. Organizers said they never intended to go to the U.S. border.

Arizona Governor Doug Ducey told a group of soldiers preparing to deploy from a Phoenix military base that their “mission is about providing manpower and resources” to support agencies on the border and denied there was a political motive.

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“I don’t think this is a partisan issue or an identity issue,” he said. “You show me somebody who is for drug cartels or human trafficking or this ammunition that’s coming over a wide-open and unprotected border here.”

Texas Governor Greg Abbott told San Antonio radio station KTSA that he would add about 300 troops a week until the total number reaches at least 1,000 troops.

Some National Guard members will be armed if they are placed in potential danger, Mr. Abbott said, adding he wanted to downplay speculation that “our National Guard is showing up with military bayonets trying to take on anybody that’s coming across the border, because that is not their role.”

There is no end date for the deployment, Mr. Abbott said: “We may be in this for the long haul.”

New Mexico Governor Susana Martinez’s office said that more than 80 troops would deploy later this week. They will be the first of an expected 250 National Guard members from New Mexico to serve on the border.

South Carolina Governor Henry McMaster, a Republican, offered to send members of his state’s National Guard as well. South Carolina sent troops to the border during Operation Jump Start, the border deployment ordered by Mr. Bush in 2006.

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Mr. Trump has said he wants to use the military at the border until progress is made on his proposed border wall, which has mostly stalled in the U.S. Congress. Defence Secretary James Mattis last Friday approved paying for up to 4,000 National Guard personnel from the Pentagon budget through the end of September.

Mexico’s Foreign Affairs Secretary said his government is evaluating its co-operation with the United States.

Luis Videgaray said in a Monday interview with local media that he will give results of the analysis to President Enrique Pena Nieto in coming weeks.

The country’s Senate passed a resolution last week saying Mexico should suspend co-operation with the United States on illegal immigration and drug trafficking in retaliation for Mr. Trump’s move.

But Mr. Videgaray said “no decision has been taken to reduce or suspend any mechanism or co-operation.”

Joel Villarreal, the mayor of Rio Grande City in Texas, said he did not agree with what he characterized as “the militarization of the border.”

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“It’s not good for business, to be frank with you,” Mr. Villarreal said.