EL PASO, Texas—When 42-year-old Angelica walked up to a U.S. Customs and Border Protection inspection booth here Wednesday, she started what is likely to be a lengthy, long shot bid to win refuge in the U.S.
Angelica and her 3-year-old granddaughter, Ruth Sofia, said they fled their home in Michoacán, along Mexico’s Pacific coast, more than a month ago amid increasing violence and threats to their safety.
As they waited before making their requests, Ruth, clad in a “Disney Princess” T-shirt, sat quietly on her grandmother’s purse. Angelica said that, if they are allowed into the U.S., they plan to reunite with her adult daughter in Modesto, Calif.
On Saturday, days after Angelica had asked for asylum, U.S. Rep. Beto O’Rourke (D., Texas) said he met with her at Annunciation House, a migrant shelter and aid group in El Paso. Mr. O’Rourke said Angelica and her granddaughter had been separated by immigration authorities.
Their requests for asylum had come shortly before President Donald Trump signed on Wednesday an executive order directing that immigrant families taken into custody at the border be kept together.
Angelica and Ruth had been among the thousands of people arriving monthly at legal U.S. border crossings and asking for the U.S. government to protect them. Tens of thousands more people, mostly from Central America, have been arrested trying to cross the border with Mexico illegally. Most of the illegal border crossers, including parents and their children and child immigrants traveling alone, are also seeking asylum.
Most won’t win, according to U.S. government data on asylum cases. About 6,300 asylum requests were approved between January and March 31, according to the Executive Office for Immigration Review, the Justice Department office that manages the immigration court system where most asylum cases are ultimately decided.
During the same period, about 90,000 people were arrested crossing the border illegally and about 32,700 immigrants went to ports of entry. Many of those people have likely asked for asylum.
According to EOIR, about 22% of the asylum cases decided this year were approved, while 41% were denied. Other cases were closed without a ruling in favor or against.
The approval rate in the court has fluctuated since 2009, the agency’s data shows. During that time, approvals rates hit a high of nearly 33% in 2011 but dropped below 17% in 2016.
Asylum seekers have been in the spotlight after the Trump administration vowed to tighten the rules around asylum and pushed Congress to overhaul immigration laws. Administration officials have referred to asylum and other laws and regulations that govern these cases as loopholes. There are also more than 700,000 cases pending in immigration court, which means cases can take years to be completed.
“Asylum and credible fear claims have skyrocketed across the board in recent years largely because individuals know they can exploit a broken system to enter the U.S., avoid removal, and remain in the country,” said Michael Bars, a spokesman for U.S. Citizenship and Immigration Services.
Mr. Sessions earlier this month ruled in one asylum case that domestic violence victims and people fleeing generalized violence, including from gangs, won’t necessarily qualify for asylum going forward.
Mr. Bars said his agency “is carefully reviewing proposed changes” with an eye toward protecting the system’s integrity, adding that Mr. Sessions’ decision will be implemented as soon as possible.
Jeremy McKinney, a Greensboro, N.C., immigration attorney, said that ruling may not have an immediate impact on the hundreds of thousands of pending asylum cases in immigration courts.
“The attorney general did not change the test for asylum,” said Mr. McKinney, a member of the executive committee of the American Immigration Lawyers Association. “Actual case law has not changed.”
The U.S. has long recognized the international right of foreigners to apply for asylum, though there is no guarantee than anyone who asks for refuge will be approved. Last year, tens of thousands of immigrants sought asylum in the U.S.
On Saturday, Mr. O’Rourke, who had been touring the Tornillo, Texas, tent shelter that houses as many as 300 child migrants, said Angelica told him she wasn’t told why Ruth was taken from her or why she was released from immigration custody.
Mr. O’Rourke, who is challenging Republican Sen. Ted Cruz in the November general election, said his office was trying to locate Ruth.
Write to Alicia A. Caldwell at Alicia.Caldwell@wsj.com