Washington (CNN) While President Donald Trump spars with lawmakers over his push to build a new border wall, the partial government shutdown is interfering with his administration's immigration agenda.

Major parts of the US immigration enforcement system have ground to a halt. And many officials tasked with carrying out immigration laws and policies are working without pay.

Here's a look at some key areas that have been affected:

Border Patrol agents are still working, but they're not getting paid

Border Patrol agents and Customs and Border Protection officers are among the tens of thousands of federal workers who are deemed "essential." That means they have to work, but won't get paid until the President and Congress can reach a budget deal.

According to a Department of Homeland Security official, about 179,000 of the agency's employees are working without pay, while 32,000 others are furloughed and won't get paid unless Congress specifically authorizes it. About 34,000 DHS employees -- the majority of whom work for US Citizenship and Immigration Services and a portion of the Federal Emergency Management Agency -- will continue to be paid because they are funded by other sources than annual appropriations, the official said.

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Representatives from the national union that represents Border Patrol agents told reporters at the White House Thursday that they're standing behind the President.

"We are all affected by the shutdown. We have skin in the game. However, it comes down to border security. And we're extremely grateful to President Trump, and we fully support what he is doing to take care of our nation's border, to take care of the future of this United States," National Border Patrol Council Vice President Art del Cueto said. "It has nothing to do with political parties. You've all gotta ask yourself this question: 'If I come to your home, do you want me to knock on the front door, or do you want me to climb through that window?'"

Many immigration courts are closed

The shutdown is yet another hurdle for an immigration court system that's already overwhelmed by a crushing backlog of cases, as the lack of funding has left many courts closed.

The Executive Office for Immigration Review, the branch of the Justice Department that runs US immigration courts, sent a furlough letter to the nation's more than 300 immigration judges on December 26.

In a notice on its website, the office said cases of detained immigrants will proceed as scheduled, but cases on the docket of immigrants who aren't detained "will be reset for a later date after funding resumes."

Already, thousands of cases have been rescheduled, according to Lauren Alder Reid, assistant director of EOIR's office of policy.

And it could be a matter of years, not a matter of a few weeks, before those rescheduled cases are heard in court, said Ashley Tabaddor, who heads the immigration judges' union.

"It's a huge, huge disruption for the orderly processing of cases," said Tabaddor, describing the shutdown as "another disruption on top of the docket shuffling the administration already does."

"It's a constant moving target," Tabaddor said, "but this is yet another huge, huge obstacle in trying to address the backlog that has been growing over the past two years despite the hiring of an additional 100 judges."

The shutdown has also interrupted efforts to hire more immigration judges and postponed the entry of a group of new judges who were scheduled to begin training on January 7, Reid said.

Jeremy McKinney, an immigration attorney in North Carolina and treasurer of the American Immigration Lawyers Association, said the shutdown is causing major problems. In one case, a hearing scheduled for Wednesday couldn't proceed because the local immigration court, which only handles non-detained cases, is closed, he said. It's not clear when it will be rescheduled.

"It's a lose-lose for the government and our clients because people, like this client from yesterday afternoon, has the anxiety and the uncertainty of potential deportation hanging over his head now for what will likely be an additional year or two," he said, adding that the shutdown also hampers the administration's deportation efforts.

"This government wants deportation to happen in a hurry," McKinney said. "That aim is being frustrated by this government shutdown."

Employers can't use the federal system to verify whether workers are in the country legally

E-Verify, the federal system that allows companies to verify whether workers are in the United States legally, isn't working due to the lapse in funding.

A statement on the program's official website states : "E-Verify and E-Verify services are currently unavailable due to a lapse in government appropriations."

That doesn't prevent new hires from starting to work. It will delay the verification process, but employees can continue to work while their status is pending.

US Citizenship and Immigration Services spokesman Michael Bars said officials have taken steps to minimize the burden on employees and employers while E-Verify is suspended.

"USCIS is committed to protecting U.S. workers, the integrity of our immigration system, and our laws," he said. "Those forcing a government shutdown in order to protest border security only hurt the interests of the people they're supposed to put first."

While the system is unavailable, employers can still -- and are required to -- submit workers' I-9 forms, which are also used to show their legal right to work in the United States. The voluntary E-Verify system goes a step further and matches the information provided by the employee on the I-9 form against federal records.

Leon Rodriguez, a former head of the agency, said the suspension of E-Verify can be particularly disruptive for employers who hire in large quantities.

"This could cause issues once E-Verify is back online," he said.

The agency should be able to handle the backup once funding is approved, he said, but it's possible systems could be overtaxed and cause further delays.

Major cases over the administration's immigration policies are on hold

Just about every major change the administration has tried to make to US immigration policies is tied up in court. And many of those cases are now on hold due to the shutdown, after Justice Department attorneys requested delays in federal courts across the country.

In at least one case, though, a federal judge said he wasn't willing to pause proceedings, despite requests from government lawyers.

US District Judge Randolph Moss, who's presiding over a case challenging the administration's asylum restrictions, said in an order last week that he wouldn't change case filing deadlines because the case is related to human safety.