WASHINGTON — When Victor and Cynthia Liu landed with their mother on a tropical Chinese island in June to visit an ailing grandfather, they thought they would soon be on a plane back to their East Coast lives — he to start his sophomore year at Georgetown University, and she to work at the consulting firm McKinsey & Company in New York.

Instead, within days, police officers detained their mother, Sandra Han, who, like her children, is an American citizen. They moved her to a secret site, commonly known as a black jail. The children discovered at the airport that they could not leave China, even though the police had said they were not being investigated or charged with a crime, the children told American officials and family associates.

By holding the family hostage, they said, the police are trying to force the siblings’ father to return to China to face criminal charges. The father, Liu Changming, a former executive at a state-owned bank, is accused of being a central player in a $1.4 billion fraud case.

The children say their father severed ties with the family in 2012, but the Chinese authorities have still held them for months under a practice known as an exit ban — a growing tactic that has become the latest flash point in the increasingly rancorous relationship between the United States and China.