Deportation officers are cutting loose more than 1,000 illegal immigrant family members a day, setting them free into border states as the surge of migrations overwhelms the government’s ability to handle them, officials revealed Thursday.

Over the last three months, about 107,000 family members were caught at the border and then released, with ankle monitoring devices or check-in schedules and the often vain hope that they will show up for their court hearings and deportation.

Worse yet, officials say, they have had to pull deportation officers off duty in prisons and jails, where they were arresting criminal migrants, and deployed them to the border to help the Border Patrol release all the families.

“The current crisis that is occurring at the southwest border, the numbers that we collectively as a nation are seeing … is absolutely unprecedented,” said Nathalie R. Asher, acting executive associate director for Immigration and Customs Enforcement’s deportation division.

As deportation officers have been pulled from their regular duties, ICE arrests have plummeted 12 percent, falling from more than 39,300 during the final three months of 2017 to just 34,546 during the same months last year.

Yet deportations actually increased, from 60,572 to 66,549.

That’s because ICE, in addition to deporting people it catches in the interior of the U.S., also deports those caught by Border Patrol agents or stopped by Customs and Border Protection officers.

And those agencies are in the middle of an unprecedented surge as children and families flock to the U.S. from Central America.

Border Patrol arrests nearly doubled, climbing from about 28,000 a month at the end of 2017 to more than 50,000 a month at the end of 2018.

The rate is still rising. Homeland Security Secretary Kirstjen Nielsen said this week that border authorities are on track to snare nearly 100,000 illegal immigrants in March.

ICE and CBP aren’t prepared for those kinds of numbers.

At the turn of the century, the overall apprehension numbers were higher, but they were almost all adult men from Mexico who could be immediately returned across the border. The current flow of children and families from Central America is more difficult. Of those caught in 2017, 98 percent are still in the U.S., officials say.

“The bandwidth is getting stretched,” Ms. Asher told reporters.

That also means ICE’s facilities are bulging.

ICE reported holding 49,960 migrants in custody on March 2, and for the fiscal year was averaging 46,766 people per day — a record level since the agency was created in 2003.

Of those detained on March 2, 34,853 were considered “mandatory” detainees based on the law.

Of those ICE did deport late last year, 66 percent had criminal convictions or faced criminal charges. The rest came to ICE’s attention because they were out of status.

By contrast, the criminal rate of deportees was 87 percent during the final months of 2015, during the Obama administration.

Under pressure from immigrant-rights groups, President Barack Obama’s team issued memos telling ICE officers not to bother with most illegal immigrants unless they had amassed significant criminal records.

President Trump’s team says it still prioritizes criminals, but there is no longer a get-out-of-jail-free card for other illegal immigrants.

Yet officials at the National ICE Council, the labor union for ICE officers, warned in a letter to Mr. Trump this month that they’re being forced to oversee a massive “catch-and-release” operation at the border.

They said ICE officers are pulled from duty on terrorism task forces and fugitive operations teams to help out at the border.

One particular gripe is that they are asked to ride along with Border Patrol agents taking illegal immigrants to bus stations, and then the ICE officers are told they have to be the ones to actually open the van doors and release the migrants. The ICE union officials said the practice appeared to be a way for the Border Patrol to say it wasn’t the agency releasing the illegal immigrants.

“Hundreds of man hours are wasted each day at a time of crisis on the border when the focus of our leadership should be streamlining efforts and eliminating redundant and unnecessary work,” the ICE Council officials said in their letter to Mr. Trump.

Deportation officers told the ICE Council they’ve been out in the field moments from arresting aggravated felons, only to be pulled out because the target was suddenly deemed “not ICE’s priority.”