Lawmakers sparred Sunday over who's most to blame for a sharp rise in migrants, including thousands of unaccompanied children, crossing into the U.S. from Mexico -- Joe Biden, or Donald Trump. 

The partisan exchanges came after the Biden administration tapped the Federal Emergency Management Agency to help lead the response. The agency, part of the Department of Homeland Security, is now working to expand available lodging for migrants.

Undocumented migration has increased significantly since President Joe Biden -- who campaigned on a pledge of offering more assistance to those seeking refugee status in the U.S. -- took office, overwhelming facilities at the border already under strain due to the coronavirus pandemic.

"It's the Biden open-border policies that are inviting even more illegal immigration and actually have created this humanitarian crisis," Texas Republican Gov. Greg Abbott said on Fox News's "Sunday Morning Futures."

In February, U.S. Customs and Border Patrol officers encountered more than 100,000 migrants attempting to enter the U.S., an almost threefold increase from a year earlier. More than 5,700 unaccompanied minors were apprehended at the border in January, the highest total for that month in recent years.

The White House has come under fire as hundreds of immigrant children are detained in rudimentary Border Patrol facilities beyond the three-day limit dictated by a 1997 court settlement. Unaccompanied minors are supposed to be transferred to the care of the Department of Health and Human Services, which has longer-term housing facilities and attempts to connect underage migrants with family already in the U.S.

"We are working in partnership with HHS to address the needs of unaccompanied children, which is made only more difficult given the protocols and restrictions required to protect the public health and the health of the children themselves," Homeland Security Secretary Alejandro Mayorkas said Saturday.

"Our goal is to ensure that unaccompanied children are transferred to HHS as quickly as possible," Mayorkas said in a statement.

The decision to use FEMA resources may help reduce the strain on some facilities, while energizing Republican critics. Mayorkas and White House press secretary Jen Psaki have refused to call the border situation a crisis while acknowledging it's become challenging.

House Speaker Nancy Pelosi said Sunday on ABC's "This Week" that Biden's administration inherited "a broken system at the border, and they are working to correct that."

But Republicans put the blame squarely on the new administration, picking up on recent comments from former President Trump, who issued a statement on March 5 warning of a "spiraling tsunami" at the border.

"The border patrol officers told me that the Biden administration's policies -- they are enriching, they are empowering the drug cartels in Mexico who make money off the people they assist in smuggling into the state of Texas," Abbott said.

House Minority Leader Kevin McCarthy is expected to lead a delegation of fellow Republicans to the border on Monday to highlight the immigration issue.

In addition to the FEMA assistance ordered Saturday, the administration in recent days announced plans to increase the number of HHS facilities where children can be transferred and to boost the number of children that can be housed in existing facilities, despite coronavirus predictions.

"It's unacceptable, but we also I think need to acknowledge that the flow of humanity arriving at our front door never stopped," Democratic Rep. Veronica Escobar, whose Texas district borders Mexico, said on CNN.

"The Donald Trump administration didn't stop them. And what we are seeing today is the consequence of four years dismantling every system in place to address this with humanity and compassion," she said, adding that migrant figures ebb and flow.

Also on Fox, Sen. John Cornyn, R-Texas, said the current trend could continue. "The August heat is deadly," he said. "In the springtime we usually see the numbers go up so I believe we're going to see an upward tick."