Three children and their mother have been released from federal custody after being detained by immigration enforcement agents last month at an upstate New York dairy farm, officials said on Monday.

The case sent shock waves through tiny Sackets Harbor, N.Y., a village of about 1,400 people on Lake Ontario where the children were enrolled, and well liked, at the town’s school.

The family’s detention was another move amid the Trump administration’s immigration crackdown to touch off protests. About 1,000 people rallied on the family’s behalf on Saturday, according to North Country Public Radio.

The release of the woman and her children — a third grader and two high school students — was announced by Gov. Kathy Hochul, a Democrat, and the local state assemblyman, Scott Gray, a Republican.

“We are open to working with federal immigration enforcement to crack down on gang members or violent criminals,” Ms. Hochul said in a statement. “But I will never support cruel actions that rip kids out of school or tear families apart.”

Mr. Gray said in a separate statement that “we are all profoundly grateful” that the family had been returned to New York.

The mother and her children, who have not been publicly identified, were taken into custody on March 27 after immigration agents arrived at North Harbor Dairy to arrest a South African man sought on a charge of distributing images of child sexual abuse, court documents said.

The man was arrested, and seven other people deemed to be in the United States illegally were detained as well, Thomas D. Homan, the Trump administration’s “border czar,” said last week in an interview with a local TV station, WWNY.

“It wasn’t a raid,” Mr. Homan said in the interview. “It was a search warrant execution at a house where a family was found in the country illegally.”

The family’s supporters disputed that characterization, saying the family had an asylum case pending in immigration court and had kept in close contact with the federal authorities.

Gittel Evangelist, a spokeswoman for the Rural & Migrant Ministry, an advocacy group in New York, said the family was from Guatemala. The New York Immigration Coalition said they had been held at a facility in Karnes County, Texas, since being detained.

The Department of Homeland Security said in a statement late Monday that the family had been detained during the execution of a criminal search warrant “to ensure these children were not being sexually exploited.”

A call to North Harbor Dairy was not returned.

School officials, who helped bring attention to the case, expressed relief about the family’s release on Monday.

“My colleagues and I are thrilled to announce that after 10 days of uncertainty, our students and their mother are returning home,” Jennifer L. Gaffney, the Sackets Harbor schools superintendent, said in a statement.

“In this difficult time,” she added, “the strength and compassion of our community have shone through to support our missing family and the entire school community.”

At the rally on Saturday, supporters of the family gathered in downtown Sackets Harbor before bringing the protest to the doorstep of Mr. Homan, marching to a vacation home he owns in the village.

Jonna St. Croix, a social studies teacher at Sackets Harbor Central School and the president of the local teachers’ union, was among those in the crowd.

The school, which runs from kindergarten through 12th grade, has about 400 students, according to state data. In such a small school, Ms. St. Croix told North Country Public Radio, everyone knew the three children.

“When there is an empty desk in the classroom, it is very evident,” she said. “And we miss them.”