In his first public statements on the deaths of two Guatemalan children in U.S. custody, President Donald Trump claimed they were "very sick" before they reached the border and foisted responsibility for their deaths on Democrats, yet both young migrants passed initial health screenings by Border Patrol.

While Trump and Democrats traded barbs over immigration policy, Homeland Security Secretary Kirstjen Nielsen was visiting medical officials and Border Patrol agents at the southern border in Arizona and Texas, promising additional wellness screenings for migrant children.

Trump, whose administration has faced widespread criticism over the deaths, pointed on Twitter at Democrats "and their pathetic immigration policies that allow people to make the long trek thinking they can enter our country illegally."

Trump's accusations came as the partial government shutdown wore on with no sign of ending over funding for sections of a border wall. Most Homeland Security employees, including Border Patrol and Immigration and Customs Enforcement, are among those federal employees required to report for work without pay.

Customs and Border Protection Commissioner Kevin McAleenan said last week that prior to this month, no child had died in their custody in more than a decade.

On Sunday McAleenan called for a "multifaceted solution" to the immigration crisis, including not only better border security and new immigration laws but providing more aid to the Central American countries from which many of the migrants have fled. This is at odds with a recent tweet from Trump threatening to cut off aid to El Salvador, Guatemala and Honduras.

"The State Department's announcement of an unprecedented increase in aid, I think, is a tremendous step forward," McAleenan said on ABC's "This Week." "There are green shoots of progress both on security and the economic front in Central America. We need to foster that and help improve the opportunities to stay at home."

In Guatemala, the mother of 8-year-old Felipe Gomez Alonzo, who died Christmas Eve, told The Associated Press that her son was healthy when he left with his father on their journey hoping to migrate to the U.S.

"When he called me, he told me he was fine. He told me not to worry because he was fine," Catarina Alonzo said from the family's home in the remote village of Yalambojoch, her stepdaughter Catarina Gomez translating her indigenous language Chuj into Spanish.

Catarina Alonzo said the last time she spoke with Felipe he was in Mexico at the U.S. border and said he was eating chicken. Their village is in Nenton municipality in Huehuetenango province, about 250 miles (400 kilometres) west of Guatemala City.

The other child, 7-year-old Jakelin Caal, died Dec. 8 in El Paso. She showed signs of sepsis, a potentially fatal condition brought on by infection, officials said.

An initial screening of Jakelin "revealed no evidence of health issues," U.S. Customs and Border Protection said Dec. 14. It wasn't until several hours later that Jakelin's father, Nery Caal, told agents she was "sick and vomiting," CBP said. Attorneys for the Caal family have also denied claims that Nery "hadn't given her water in days," as Trump wrote.

And CBP said Tuesday that agents checked on the well-being of Felipe and his father 23 times in the first several days the two were detained. Felipe's father, Agustin Gomez, told a Guatemalan official the boy first showed signs of illness the morning of the day he died.

Despite Trump's claim that Democrats were responsible for "pathetic" immigration policies, at least one of the laws his administration has blamed -- legislation that prevents the immediate deportation of unaccompanied children from Central American countries -- was signed in 2008 by President George W. Bush, a Republican.

Democrats lambasted the president's tweets. In a tweet addressing the president, Sen. Mazie Hirono wrote: "Obviously nothing is too low or cruel for you. A collective New Year's wish: For the sake of our country, you can stop now."

"You slander Jakelin's memory and re-traumatize her family by spreading lies about why she died," said U.S. Rep. Joaquin Castro of Texas.

Meanwhile, Nielsen travelled to Yuma, Arizona, on Saturday to meet with medical staff at the border. She said in a statement that "the system is clearly overwhelmed and we must work together to address this humanitarian crisis" and she called on Congress to "act with urgency."

She was also briefed in El Paso, Texas, on Friday on "recently instituted secondary medical screenings and the more thorough initial health screenings of migrants."

El Paso Mayor Dee Margo, a Republican, said he met with Nielsen and told CNN on Saturday that he agreed with her that immigration policy is "broken."

Felipe and Agustin Gomez were apprehended by border agents Dec. 18 near the Paso del Norte bridge connecting El Paso to Juarez, Mexico, according to border officials. The two were detained at the bridge's processing centre and then the Border Patrol station in El Paso. They were transferred Dec. 23 to a facility in Alamogordo, New Mexico, about 90 miles (145 kilometres) away.

After an agent noticed Felipe coughing, father and son were taken to an Alamogordo hospital, where Felipe was diagnosed with a common cold and found to have a fever of 103 degrees Fahrenheit (39.4 degrees Celsius), officials have said.

Felipe was held for observation for 90 minutes, according to CBP, before being released with prescriptions for amoxicillin and ibuprofen.

But the boy fell sick hours later on Dec. 24 and was re-admitted to the hospital. He died just before midnight.

New Mexico authorities said late Thursday that an autopsy showed Felipe had the flu, but more tests need to be done before a cause of death can be determined.

The government of El Salvador is pushing back against Trump's assertion it doesn't do enough to stem migration to the United States. In a statement released Saturday, the government says it has made strides in economic and social improvements to try to tamp down the root causes of the phenomenon, and has pushed a media campaign urging its citizens not to risk their lives making the dangerous journey. It says migration from the country has fallen significantly this year.

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Merchant reported from Houston and Miller and Long from Washington. Marcos Aleman in El Salvador and Sonia Perez D. in Guatemala contributed to this report.

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This story has been corrected to show that 8-year-old Felipe Gomez Alonzo died in Alamogordo, New Mexico, not El Paso, Texas.