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 & Co. in New York.

Instead, within days, police officers detained their mother, Sandra Han, who, like her children, is a U.S. 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 police had said they were not being investigated or charged with a crime, the children told U.S. officials and family associates.

By holding the family hostage, they said, 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 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.

Senior U.S. diplomats, already contending with tensions over trade and territorial disputes, have denounced the way China uses exit bans as coercive, opaque and a violation of rights. In January, the State Department issued a travel warning, saying the practice posed risks to foreigners in China. Last month, Secretary of State Mike Pompeo criticized the bans during a visit to China, and this month he mentioned the Liu family to a top Chinese foreign policy official, Yang Jiechi, at a meeting in Washington.

The siblings have pleaded their case to U.S. officials, including John Bolton, the national security adviser. “The investigative officers have made abundantly clear that neither my brother nor I am under any form of investigation,” Cynthia Liu, 27, wrote to Bolton in an August letter obtained by The New York Times. “We are being held here as a crude form of human collateral to induce someone with whom I have no contact to return to China for reasons with which I am entirely unfamiliar.”

She made similar points in an email sent in recent days to a family associate, saying that “the Chinese authorities have been consistent that neither Victor nor I are accused of or suspected of any criminal activity,” and that authorities have repeatedly said that “the reason we are here is exclusively to lure” their father.

On Friday, in response to questions from The Times, a State Department spokesman, Robert Palladino, said the United States would continue expressing concern about exit bans “until we see a transparent and fair process.”

The Chinese Foreign Ministry defended the holding of the three family members, saying: “The people you mentioned all own legal and valid identity documents as Chinese citizens. Because they are suspected of economic crimes, they are restricted from exiting the country by the Chinese police in accordance with the law.”

This account of the Liu family’s plight is based on interviews with administration and congressional officials, university employees and family associates, as well as correspondence by the children to them and public records. There are currently a handful of exit-ban cases involving Americans, but this is the first one in which those with knowledge of it are making the details public.

It is also a rare instance in which one of those being held — Victor Liu, 19 — was born in the United States. More often, China imposes exit bans, which can last from days to years, on naturalized foreign citizens who were born in China. Security officers often treat them as if they were still Chinese citizens, even though China does not recognize dual nationality.

The law in China states that its nationals automatically lose their citizenship when they gain citizenship in another country. It also says that someone like Victor Liu who is born with citizenship from another nation is not a Chinese citizen, no matter the parents’ citizenship.

The Liu family members all entered China on U.S. passports, and the State Department is providing them with citizen services. Police from Guangzhou, the southern provincial capital where the father worked, have taken Han, 51, to a hotel several times to meet with a U.S. consular officer.

Liu Changming, 53, the father, is among China’s most-wanted fugitives, accused of helping to carry out one of the country’s biggest bank frauds, in which $1.4 billion in illegal loans was issued to property developers. He fled the country in 2007.

In recent years, as Chinese authorities have waged a fierce crackdown on corruption, they have increasingly viewed family members of corrupt officials as being tied to the illegal acts, with police sometimes aiming to prosecute spouses.

The Liu family has lived with notable wealth in the United States, in part by acquiring real estate, though there is no indication from public records that the properties were purchased with the money that the father is accused of embezzling.

Sen. Elizabeth Warren, Sen. Ed Markey and Rep. Joseph Kennedy III, D-Mass., where the mother lives, have urged the U.S. government to act.

“Our office is aware of Victor’s, Cynthia’s and Sandra’s situations and is deeply concerned,” Markey’s office said in a statement. “We are working to secure their safe return and continue to be in touch with U.S. officials to ensure a positive outcome.”

— Purgatory With Police

The Liu children are trapped in a situation far from their elite American lives. Both attended Groton, an expensive Massachusetts boarding school. Cynthia Liu graduated from Stanford and Harvard Business School. The family has a $2.3 million house in a Boston suburb, and the mother, a businesswoman, controls or has her name on companies with real estate holdings worth at least $10 million, including two luxury apartments in Manhattan, according to public records. Since June, the children have tried to leave China three times. They travel between cities, staying in hotels or with an uncle. Their ailing grandfather died in October. They limit their electronic communications because of suspected surveillance.

“Out of concern for the security of these young Americans, we will refrain from public comment as we continue our efforts to constructively and directly engage the Chinese government to allow them to return home,” said David Pressman, a partner at the law firm Boies Schiller Flexner who is representing the Liu children.

The Georgetown president met with the children this month in Beijing. Harvard has written to the Chinese ambassador in Washington. McKinsey is in touch with Cynthia Liu. People tracking the case say they hope President Donald Trump will raise the issue at the G-20 summit meeting this week in Argentina, where he and Xi Jinping, the Chinese president, are expected to meet.

China has convicted foreigners of crimes and even executed them. It has also detained Muslims who are foreign citizens in its northwest internment camp system. The exit bans are different — authorities do not keep the people at a detention site or present a legal justification. They use the ban mainly as a means of coercion: to squeeze out information, to compel a relative or friend to return to China, or to force the victim to settle a business dispute. In one case, an American stayed in China under an exit ban for more than two years. A continuing case involves a Singaporean national working for UBS, the Swiss bank, which has prompted UBS to issue travel warnings to its employees.

“Many police actions are now taken outside the legal system, and both Chinese citizens and foreigners, especially those who used to be Chinese citizens, have been kept in China with no apparent legal authority,” said Jerome A. Cohen, a law professor at New York University.

China’s increasing use of the practice is a result of major trends: a growing disregard for civil rights among Chinese security forces; a sweeping anti-corruption campaign started by Xi in 2012; and a rush among corrupt officials and executives to move overseas with their families. In 2014, China announced the start of a global campaign to hunt down fugitive former officials.

Many of the former officials live overseas in luxury, with new names and citizenship. China has sent secret agents to the United States to try to retrieve some. China has also asked the United States to send back former officials, but the two countries do not have an extradition treaty.

U.S. officials have been reluctant to cooperate because of China’s human rights abuses and lack of rule of law, though there have been exceptions — including the repatriation of a former vice mayor accused of stealing $39 million. — The ‘Naked Official’

In 2007, Chinese auditors closed in on what was becoming the country’s biggest bank fraud case. Liu Changming fled China that December, state media reported. The top official at the Guangzhou branch of the Bank of Communications, he was at the center of the case.

Officials charged him in 2008 with making illegal loans of about 9.8 billion yuan, or $1.4 billion, including some to a company he secretly controlled, according to Caixin, a financial newsmagazine. Officials convicted Liu’s co-conspirators but recovered only half of the money.

In 2015, China put Liu on its “Skynet” list of 100 most-wanted fugitives. Interpol issued a “red notice” for his arrest. His whereabouts are unknown.

The Financial Times reported in 2009 that after escaping China, Liu took part in shareholder meetings in London for Canton Property Investment Ltd., a company whose Chinese subsidiaries received the illegal loans. The company had gone public in London in August 2007 and raised $50 million, but was delisted the next year.

Public records show that a person named Changming Liu is linked to a home in Chestnut Hill, Massachusetts. The address matches the one Cynthia Liu lists on her Harvard alumni page. The home is owned by Sandra Han, the mother. Zillow, the real estate website, estimates its value at $2.3 million.

A company controlled by Han bought the home in 2009. Several real estate companies, trusts and limited liability corporations are registered to that address, and they in turn own rental properties in Massachusetts and luxury apartments in New York.

While Liu was moving up the ranks of China’s banking bureaucracy, he became what the Chinese call a “naked official” — someone who settles his family abroad, with relatively easy paths to foreign citizenship and well-regarded schools, and with Chinese authorities an ocean away.

The mother and daughter moved to California in 1998, and by early the next year they lived in a condominium in Alhambra, a Los Angeles suburb popular with Chinese immigrants. In June 1999, Liu and his wife bought a three-bedroom home in nearby Arcadia. Records show they obtained Social Security numbers in California around this time.

Their son Victor was born in California that July. He later lived in southern Chinese cities for five years, leaving Guangzhou in 2007.

The couple sold the Arcadia home in 2004 and bought a house in a gated community in Armonk, New York, an affluent town, records show. The home was transferred to Cynthia Liu’s name in 2011 and sold in 2014 for almost $900,000.

Han is a trustee of an entity called Bountiful Success Trust — 2014, which bought a $2.77 million condominium in downtown Manhattan in 2016. Another company she controls owns a unit at the Lucida on the Upper East Side that Zillow values at $4.17 million.

Like his mother and sister, Victor Liu is interested in business, and he rented a room this summer with Noor Darwish, his first-year roommate, in a Georgetown home to do an internship at his university’s endowment office. But he left suddenly in June. He called Darwish as he was leaving the country.

“He told me he had some family problems, and that his grandfather had had a heart attack,” Darwish said.

The university has strongly pushed for the siblings’ release. On his trip to China this month, the Georgetown president, John J. DeGioia, met with senior officials. With him was Evan S. Medeiros, a professor who had helped negotiate releases of U.S. hostages in North Korea as senior Asia director in President Barack Obama’s National Security Council.

The university has a little-known connection to China’s leadership. A nephew of Xi, China’s president, is a recent graduate of Georgetown’s School of Foreign Service. While at Georgetown, he also interned at the Brookings Institution.

None of that has helped. With winter break looming, Liu’s friends on campus are asking why he has been absent all semester. “Our room is a lot lonelier now,” Darwish said.

In China, Liu has been eating and sleeping poorly. His sister mentioned concerns about her own health in her letter to the White House. She wrote, “We feel alone, angry, and most of all afraid, and more than anything, we just want to come home.”