Lisa Miller, the girl’s biological mother and a newly fervent Baptist, was championed by conservatives for her efforts to shield her daughter from homosexuality. A Vermont court had granted her primary custody of the daughter, Isabella Ruth Miller-Jenkins, after Ms. Miller split with her partner, Janet Jenkins, in 2003. But the court also declared Ms. Jenkins to be a legal parent with liberal visiting rights, and Ms. Miller, who had moved with the girl to Virginia, defied repeated orders to permit the visits.

Image Janet Jenkins has legal custody of a daughter born with Lisa Miller. Credit... Lisa Billings/Associated Press

The case took a turn in late 2009, as the Vermont family court, citing Ms. Miller’s noncompliance, shifted primary custody to Ms. Jenkins. Ms. Miller and Isabella, who is now 9, disappeared. A warrant was issued for Ms. Miller’s arrest, and they have not been heard from since.

According to an F.B.I. affidavit unsealed in Vermont on Thursday, the pastor, Timothy David Miller of Crossville, Tenn., helped arrange in September 2009 for Ms. Miller and Isabella to fly from Canada to Mexico and travel on to Nicaragua, where he worked as a missionary for Christian Aid Ministries. (The F.B.I. said it had no evidence that Mr. Miller and Lisa Miller were related.)

Ms. Miller and Isabella stayed in a beach house in Nicaragua that is owned by a conservative businessman whose daughter works at the law school at Liberty University, an evangelical school in Lynchburg, Va., according to the affidavit.

Lawyers from Liberty, including the dean of the law school, Mathew D. Staver, represented Ms. Miller in court appeals on the custody issues. They argued without success that Ms. Jenkins had no parental rights and that laws in Virginia, which ban same-sex unions, should prevail over those in Vermont.