Santos Lopez uprooted his family and walked nearly 2,000 miles on a dangerous trek from Honduras to the United States last spring to escape from a violent gang that was extorting him. The group demanded a monthly payment, he said, to allow him to run his car shop in peace.

Like many others, Mr. Lopez and his family hoped their experience would persuade their adopted country to give them asylum, which is granted to those who face a “credible fear of persecution in their country of origin.” A grant of asylum would allow them to work and eventually apply for a green card and citizenship.

But more than a year after his family — including his wife and two daughters — arrived safely at the southern border, it seems likely they missed the deadline to apply. Mr. Lopez, 42, said he was seeking help from a lawyer.

Mr. Lopez and his family are among the millions of migrants who have arrived at the southern border in the past year. Many, after telling border agents about abuse and persecution that they experienced, a first step in the long and complicated process of seeking asylum, have been temporarily released as they wait for their immigration cases to wind their way through courts.

But even as migrants have applied for asylum in record numbers, advocates and immigration attorneys say that without additional legal support, many — perhaps the majority — will miss their application deadline and fall into a more perilous category of immigrant: the undocumented.

“Our immigration system is broken,” said Henry Love, vice president for policy and advocacy at Win, which runs 14 family shelters and has a contract with New York City to house migrant families.

“You’re going to have so many people who won’t have the opportunity to apply for asylum simply because of the logistical complications of it,” he said, adding: “I have a Ph.D., and there’s no way I could do it.”