In the case of Ms. Merezhko, who has lived in the United States since 2014, the family had made a decision to try to get Ivan to safety in Los Angeles as quickly as possible, without waiting for the United States to begin issuing permission for refugees to fly directly. Entering through Mexico, which does not require visas for Ukrainians, has been a stopgap measure for an estimated 5,000 Ukrainians since the war began in February.

Ms. Merezhko spent about $7,000 to purchase airline tickets, took a leave from her job as a pharmacy technician and set out to retrieve the boy. They rendezvoused in western Ukraine, after he had managed to board an evacuation train out of the besieged city of Kharkiv, where the family lived.

“I thought I was doing the right thing because it was the only way to save the child, to bring him to a safe place,” she said.

From Madrid, they boarded a flight to Monterrey, Mexico, and connected on April 6 to Tijuana. They slept in a tent erected outside a gym that was already overflowing with Ukrainians waiting to report to the border checkpoint for processing.

When it was their turn two days later, Ms. Merezhko said, Customs and Border Protection officers carefully studied the documents and notarized letter from Ivan’s mother stating that her sister had been granted full responsibility for him.

An officer told them that they would have to be separated — for just one or two days.

“Everything will be OK,” Ms. Merezhko assured Ivan.

The following day, her phone rang, and an officer put Ivan on the line. “Mama, Mama, is that you?” he asked, thinking it would be his mother in Ukraine.

His aunt’s heart sank. In the 60-second exchange, all that he was allotted, the boy said he was still at the border. The officer told Ms. Merezhko to expect another call soon.

Days went by, no call came and her anxiety mounted.

Ms. Merezhko learned that Ivan was now probably at a government shelter, and she found the number of a hotline for families trying to locate children.

An attendant confirmed that Ivan was in the system, and told his aunt that she would be contacted by a case manager in a few days.

“I got no information about how he is, where he is,” Ms. Merezhkho recalled.

Days passed.

She called the hotline again, and an operator urged her to be patient: It could take 20 to 30 days before Ivan was released, she said, and that process had not even started.

During an anguished call, Ms. Merezhko’s sister, Kateryna, told her that she now regretted sending away her only child. “At least we would know where he was if he had stayed with us,” she told her.

Over the weekend, with the help of Ms. Revkin, from the nonprofit, Ms. Merezhko filled out 25 pages of forms to which she attached green cards, marriage certificates, birth certificates and pay stubs for her and her husband. Yet she had nowhere to send the file.

Finally, on Monday, Ms. Merezhko’s husband, Vadym, got a call from Ivan, who said he was at a shelter in California. A case worker said they could send the documents. But there still was no word on when Ivan would be released.

“Good news today,” Ms. Merezhko said. “But I am a little bit worried about how long it will be.”

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