Immigration officers told them the next day that they would be released if they provided contact information for a family member in the United States who could become their sponsor, she said. But after they gave the name of a relative, she wrote, immigration officials said there was a change in plans.

They were turned over to Immigration and Customs Enforcement, wrote Ms. Connors, 24. Mr. Connors, 30, was dropped off at the Tacoma Northwest Detention Center, while she and her son were taken to a Red Roof Inn in Seattle. They were in customs custody for a much shorter period then most border crossers, Ms. Cambria said. “They had an understanding of the language and their rights and were outspoken,” she said, adding that many others do not have the same luxury.

The next morning they were put on a cross-country flight to Pennsylvania, where they arrived at the Berks Family Residential Center in rural Leesport, she wrote. The other adults and their infant twins were taken to the same place, Ms. Cambria said.

The baby had not completed his immunizations, making him especially vulnerable to germs and contagious diseases, Ms. Cambria wrote in a letter to Homeland Security, adding that he had not yet seen a pediatrician.

“We all had to shower when we arrived, but because he is too small to take into a shower, I had to wash him off in the office on a sofa, using a washcloth and soap that I had with me,” Ms. Connors wrote in her sworn statement. “They offered me a baby bathtub to use, but it was filthy dirty and had broken bits, so I refused to use it.”

She wrote that the temperature dropped to 50 degrees at night, and that the staff refused to turn the heater on. On Friday, the baby woke up with a swollen, teary eye, and his skin was rough and blotchy, she wrote. Officials did offer to remove the baby, she wrote: “If we wanted, we could sign papers to allow him to be separated from us and taken to some other facility.”

Immigration and Customs Enforcement officials did not return a request for comment. About 4,300 people were arrested at the Canadian border in 2018, and about 3,000 in 2017.