The United States on Thursday called on Qatar, the tiny gulf emirate and important Arab ally, to lift a travel ban on an American couple held there for more than 19 months on a criminal prosecution over the death of their adopted African daughter, a case that critics contend is riddled with irregularities, racial prejudice and cultural misunderstandings.

In a statement, the State Department said it was concerned about the well-being of the couple, Matthew and Grace Huang of Los Angeles, and their two sons, who were both returned to the United States in the custody of other relatives after the parents were arrested and imprisoned in January 2013.

The State Department said it had requested the Qatari government’s “assistance in providing a fair and expeditious conclusion to the proceedings,” which have moved slowly even though the original charge of murder was thrown out, replaced by the lesser charge of child endangerment and a conviction in March that carries a three-year prison term.

The Huangs, who assert they are innocent of any wrongdoing and have appealed, are not incarcerated while the appeal is pending but are prohibited from leaving Qatar. The next court hearing is in October. Prosecutors are seeking a longer sentence.