McALLEN, Texas >> At the Republican debate Wednesday and throughout the campaign, candidates led by Donald J. Trump have assailed illegal immigration, and some have questioned whether children who are born to immigrants in this country illegally should be considered U.S. citizens.

But here on the Texas border some communities are engaged in activities that go beyond talk, enforcing some of the toughest rules in the country limiting the types of identification parents can show to receive copies of birth certificates. The result has been a refusal to issue birth certificates to many of the Texas-born children of immigrants here illegally.

The state says it is a matter of adequate verification. Organizations that advocate on immigrants’ behalf see anti-immigrant sentiments turned into anti-immigrant policies — with the goal of making it harder for unauthorized immigrants to live and raise families in Texas.

The requirements have spurred a lawsuit against the state and opposition from Hispanic lawmakers, immigrant rights activists and even officials from Mexico.

Without a birth certificate, immigrants say they cannot have newborns baptized, have had difficulty enrolling their children in day care and school and have lost or fear losing Medicaid coverage and other government services and benefits for their families. They said officials required birth certificates for these programs, as proof of parenthood or the child’s Texas birth.

Some have older children who have birth certificates but younger children who do not, because local registrars who used to accept certain types of foreign-issued ID no longer do.

"It’s not right," said a 34-year-old woman who was denied birth certificates for two of her children in McAllen, and who asked that her name not be used because she feared reprisals from the authorities. "Yes, I’m here illegally. But I’m the one who committed the crime, not them."

In court papers, lawyers for Texas disputed some of the claims made by those denied birth certificates, arguing that not having a copy of a birth certificate does not prevent a child from attending school or qualifying for Medicaid. And they say state policies do not discriminate against immigrants but are meant to insure that birth records do not fall into the wrong hands.

"Vital records contain private information that is confidential by law," said Chris Van Deusen, a spokesman for the Texas Department of State Health Services, which oversees the state’s Vital Statistics Unit. "State and local registrars have a duty to protect that information by ensuring they only release records to people who are qualified to obtain them."

The 14th Amendment grants citizenship to anyone born in the United States, and Texas has not directly challenged the citizenship status of the children. Many of them have birth certificates in the state’s database.

Most Texans pick up a birth certificate by showing a driver’s license. But immigrants in the country illegally lack many of the approved IDs, such as a license or foreign passport with a U.S. visa.

Unauthorized immigrants and their lawyers said local Texas registrars used to accept an ID card with a photograph issued to the immigrants by foreign consulates. The matricula, as the Mexican version of the consular ID is known, has been used by immigrants to open bank accounts in Texas and receive other services.

Although the official Texas policy prohibited the use of consular IDs to obtain birth certificates, the prohibition was not strictly enforced until about 2013, when state officials in Austin began pressing local records clerks to follow the protocols, immigration lawyers said.

"People in Austin started tightening the screws," said Efren C. Olivares, a lawyer with the South Texas Civil Rights Project who represents the parents in the lawsuit.

The refusal to accept the consular IDs coincided with the tough-on-immigration stance of Texas Republican leaders, the influx of Central American immigrants at the border last summer and the state’s opposition to President Barack Obama’s executive actions on immigration, which would give temporary reprieves from deportation to as many as 4 million immigrants and also permit them to work.

Unauthorized immigrants and their lawyers said Texas officials had been intentionally requiring IDs unavailable to them as part of their efforts to bolster border security. One couple denied a birth certificate for their 6-month-old daughter say in court documents that they were told by a McAllen official that the requirements became stricter to prevent immigrants from obtaining legal status through their U.S.-born children.

Obama’s executive actions on immigration, announced in November, would offer three-year deportation deferrals and work permits to millions of unauthorized immigrants who have a child who is a U.S. citizen or legal permanent resident, and who have been living in the United States for five years and have no serious criminal records.

Texas officials say that immigration status plays no role in determining who is supplied a birth certificate. They say that the consular ID cards are not secure because the consulate offices do not verify the documents presented to obtain them, and that immigrants can use other types of identification to receive copies of birth certificates. The list includes student IDs, Medicaid cards, Mexican voter registrations, utility bills and paycheck stubs.

Van Deusen of the State Health Services Department said the agency had never accepted consular IDs, but he suggested that local registrars had allowed them in the past without the state’s approval.

"We can’t speak to what the practices of each of the more than 450 local registrars have been over the years," he said in a statement. "If we’ve encountered a local registrar that was accepting consular IDs, we’ve asked him or her to stop and explained why."

Unauthorized immigrants in other states do not appear to have had similar problems with birth certificates.

New Mexico accepts the consular ID cards that Texas prohibits. California officials said they did not have any policies on the use of consular IDs to obtain birth certificates. Although Arizona does not accept consular IDs, officials there said they work with those who lack valid ID, including allowing witnesses who know the parent and who have proper ID to take part in the application process. In Nevada, one of the accepted IDs is a hospital birth record, which many of the Texas families have but the state does not accept.

"There’s no other state that has locked people out the way Texas has," said Jennifer K. Harbury, a lawyer with Texas Rio Grande Legal Aid who is representing the children in the lawsuit.

Plaintiffs in the suit are about two dozen families from Mexico, Honduras and Guatemala. The number has grown since the suit was first filed in May and now includes 28 adults and 32 children. The lawyers for the families said there were hundreds and possibly thousands of others who had been denied birth certificates.

"We’ve had people who’ve come to us asking for help, but they didn’t want to join the lawsuit because they’re afraid," Olivares said. "They don’t want the potential repercussions of coming out publicly as undocumented."

One 40-year-old unauthorized immigrant who was denied a birth certificate for her 9-month-old daughter said she could not work as a result: Day care centers want a birth certificate, so she remains unemployed because she has no one to watch the baby.

She and other immigrants said they now limit their travels because they lack a birth certificate. If they drive north, they have to pass inland Border Patrol checkpoints, and they fear being separated from their children because they lack proof they are the parents.

Mexico filed a brief in support of the parents suing Texas, with its lawyers calling the state’s actions a violation of international law that introduces "a troublesome and discordant element into binational or transnational relations."

Gov. Greg Abbott expected to discuss the issue with Mexican officials this month during his visit to Mexico City, but the topic did not come up, a spokesman for the governor said.

"One thing both sides understand is that the Texas-Mexico relationship is bigger than any one issue," said the spokesman, John Wittman.