For Yessica Rojas, the choice was clear. After less than two years in Missouri, she and her two children had to leave.

The reason, she said, were stories about Venezuelan mothers like her that had gone viral on social media. Ms. Rojas heard they had been deported to Venezuela while the American authorities held on to their children.

“It just isn’t worth it now,” said Ms. Rojas, 29, explaining her decision to leave the United States.

The stories are not just rumors: This year, a Venezuelan 2-year-old called Antonella by her family remained in foster care while her mother was sent back to Venezuela and her father to a prison in El Salvador. (Antonella has since been returned to her family.)

Ms. Rojas left Branson, Mo., the same week.

Episodes like this one have sent shock waves through immigrant communities, accomplishing what the Trump administration’s warnings — and even the promise of $1,000 for those willing to “self-deport” — alone could not do: They have persuaded at least some migrant parents to leave the United States.

It is not clear how many people have abandoned their American lives since President Trump began his second term. A recent flight carried around 65 people who opted to leave for Colombia and Honduras, according to the Trump administration, and individual stories of immigrants leaving voluntarily have emerged across the country.