LOS ANGELES — Marlon Parada, a construction worker in Los Angeles, already was worried when he got an urgent call from his cousin in Honduras, asking if he would agree to take in the cousin’s 14-year-old daughter. She’d been taken from her mother while attempting to cross the border and detained in Houston, he said. She couldn’t be released unless a family member agreed to take her in.

Mr. Parada, an immigrant himself who is supporting his wife and three daughters on $3,000 a month, wondered how he could afford to take on another responsibility. Then he learned that he would have to pay $1,800 to fly Anyi and an escort from Houston to Los Angeles.

“It caught me by surprise when they demanded all that money. I asked them to just put her on a bus, but they wouldn’t,” said Mr. Parada, who scrambled to amass the cash from friends and wired it to the operator of the migrant shelter where Anyi was being held.

But that was only one of the hurdles he would have to surmount to take custody of the girl. Families hoping to win release for the thousands of migrant children being held by federal immigration authorities are finding they have to navigate an exhausting, intimidating — and sometimes expensive — thicket of requirements before the youngsters can be released.