The US government has released more than 1,600 migrants on to the streets of El Paso, Texas, this week, overwhelming aid agencies that have scrambled to find shelter for families left to fend for themselves.

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Advocacy groups said the mass release was “unprecedented”, as volunteers turned out in droves to bring food, water and medicine to migrants stranded in the border city.

“It is unacceptable to release families with children, some of them very young children, to the street,” said Ruben Garcia, executive director of Annunciation House, El Paso’s largest migrant shelter operator. “That, we have to say, is a non-negotiable, that is simply not acceptable. Whatever logistics, whatever pressures may exist, we have got to find a system to ensure families are not released to the street.”

The chaos began last Sunday night, when 214 people, all families, were released without the usual warning given by Immigration and Customs Enforcement (Ice) to Annunciation House, Garcia said. Volunteers worked until at least 2am or 3am trying to find shelter, Garcia said.

Hundreds of people were then released on Christmas Eve, Christmas Day and Thursday.

Garcia said immigration authorities had coordinated travel for released families with Annunciation House and its network of shelters for “the last five years”, until this week.

“I cannot explain to you why that was done,” he said.

Garcia said Annunciation House and its local partners have provided space to about 2,200 people a week, and are hoping to expand to at least 3,000 beds a week. The group is spending $150,000 a month for hotel rooms because its shelters are full. On Friday, the El Paso Community Foundation and the Prudential Foundation gave $40,000 to Annunciation House to help pay housing costs.

El Paso’s incoming US representative, Veronica Escobar, a Democrat, told CBS News the situation could be exacerbated by an annual college football game, the Sun Bowl, taking place this weekend, because it will bring in tourists who have booked hotel space.

She called on the government to build temporary holding facilities and to invest in programs that could help address the poverty and violence driving people to flee from Central America.

“As a community, we are going to do whatever we need to do to take care of these people,” Escobar said. “They are vulnerable. They deserve compassion. And help. And support. And El Pasoans always rise to that occasion.”

Ice, which must take custody of families within 72 hours of their apprehension by border patrol, said it does not have the capacity to detain or remove families quickly and has cut back reviews of post-release plans for migrants held at the border.

In an email, a spokeswoman said: “Ice is redoubling its efforts to work with local and state officials and NGO partners in the area so they are prepared to provide assistance with transportation or other services.”

Facebook Twitter Pinterest Central American migrants, who were dropped off at a bus station by Ice, walk toward volunteers helping them with food and transportation to emergency shelters in El Paso, Texas, on Christmas Day. Photograph: José Luis González/Reuters

Across the border in Ciudad Juárez, the city’s biggest migrant shelter, Casa del Migrante, said on Christmas Eve it would no longer be able to accept new migrants. Javier Calvillo, director of Casa del Migrante, said the facility only had enough food, water and medicine to care for the nearly 500 people it was already housing.

“I think it is important to recognize the limitations, what we have and what we can do,” Calvillo said.

In recent months, border towns have been overwhelmed by Trump administration efforts to restrict asylum while thousands of Central Americans continue to arrive. At some of the busiest ports of entry, journalists and humanitarian groups have documented the US government limiting the number of people who can apply for asylum each day, a practice known as “metering”.

This month, the supreme court confirmed people can seek asylum anywhere along the US-Mexico border, despite White House attempts to prohibit the practice.

Such efforts to curb legal and illegal immigration come as apprehensions of people illegally crossing the border are at their lowest levels since the peak of 1.6 million people in 2000. As of October, 396,579 people had been apprehended this year.

The homeland security secretary, Kirstjen Nielsen, visited El Paso on Friday, after being pressured by lawmakers to provide more information on the circumstances surrounding the deaths of two Guatemalan children in US Customs and Border Protection custody this month.

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Jakelin Caal, seven, died on 8 December, less than two days after being apprehended. Her cause of death is not known. Felipe Gómez Alonzo, eight, died late on Christmas Eve. New Mexico authorities said an autopsy showed he had flu, but more tests were needed.

Nielsen did not immediately address the situation in El Paso, but did acknowledge that the immigration system was broken.

“The system is clearly overwhelmed and we must work together to address this humanitarian crisis and protect vulnerable populations,” Nielsen said in a statement on Saturday, minutes after Trump blamed the children’s deaths on Democrats holding out against his demands to fund a border wall (for which, on the campaign trail, he had promised that Mexico would pay). The Department of Homeland Security has also said it will hold more thorough health screenings for migrants.

Bishop Mark Seitz, of the Catholic diocese of El Paso, said he understood government facilities were overcrowded but turning people on to the street without money, food or access to phones was “inhumane and unconscionable”.

“Our government has a responsibility when it takes these refugees into custody to provide for their basic needs. We do as much for death row inmates,” Seitz said in a statement.

“With the recent deaths of two young children, the accounts of mistreatment in holding cells and the heartless expulsion of these families an undeniable picture is coming into focus and it is not one of which we Americans can be proud.”