Jean Jacques was fresh out of prison for attempted murder in 2012, and immigration agents wanted to deport him back home to Haiti — but the island nation’s government refused to take him back, saying it couldn’t be sure Jacques was who he claimed to be.

At that point U.S. Immigration and Customs Enforcement should have enlisted the State Department, then run by Secretary Hillary Clinton, to get involved and put pressure on Haiti, but agents didn’t do that. They would later explain to investigators they’d learned from experience that the State Department didn’t get involved in cases as low-level as Jacques‘ attempted murder.

Several years, and repeated failed deportation attempts later, Jacques killed a 25-year-old Connecticut woman, Casey Chadwick, stabbing her to death in a drug dispute with her boyfriend, prosecutors said.

He is one of thousands of criminal immigrants who went free during Mrs. Clinton’s time in office, released from custody because their home countries wouldn’t take them back.

This week GOP presidential nominee Donald Trump laid the death of Chadwick and others squarely at the feet of Mrs. Clinton, accusing her of failing American families by not using the tools she had to force other countries to comply, such as halting the issuance of U.S. visas to their own citizens.

“She had the power and the duty to stop it cold, and she decided she would not do it,” Mr. Trump said Wednesday as he laid out his immigration policy, combining a get-tough approach with a fierce critique of the way both Republican and Democratic presidents have handled the issue.

Indeed, the State Department’s reluctance to get involved in these disputes predates Mrs. Clinton’s tenure. For years members of Congress have begged the department to get tough on countries that refuse to take back their citizens.

But first the Bush administration, and now the Obama administration, have balked, saying diplomatic relations with some of the worst offenders are too complex to be ruined by immigration disputes.

“The problem of countries not taking back their deportees had been around for some time, but during her time as secretary of state, Hillary Clinton did nothing meaningful to address it, did not make it a priority and failed to utilize the tools in the law that would have helped,” said Jessica Vaughan, policy studies director at the Center for Immigration Studies.

Some 23 countries are currently deemed “recalcitrant,” meaning they habitually refuse to accept their own citizens. Haiti isn’t one of them, with officials saying, despite the Jacques case, that the Caribbean nation is usually cooperative. But China, Cuba, Guinea and Liberia top the naughty list.

Government documents show 35,000 criminals from Cuba are running free in the U.S. because that country won’t cooperate. China’s deportation backlog, meanwhile, stands at 1,900.

Under federal law, the government is supposed to stop issuing visas to countries that won’t cooperate in deportations. The Homeland Security secretary is supposed to issue a notification, but the State Department can also act on its own.

The power has been used just once, in 2001, with Guyana, and it was spectacularly successful. Within two months 112 of the 113 immigrants in the backlog to that country had been deported.

Calls to exercise the power under Mrs. Clinton and her predecessor, Condoleezza Rice, and her successor, John Kerry, have gone unheeded.

State Department officials didn’t respond to a request for comment. Neither did Mrs. Clinton’s presidential campaign.

For his part, Mr. Trump said he’d force other countries to take back their deportees as part of a 10-point get-tough immigration plan that also included stepping up deportations, enforcing existing laws and targeting criminals.

During Mrs. Clinton’s tenure, the State Department signed a memorandum of understanding with Homeland Security that laid out a series of other steps that could be taken, including joint diplomatic meetings and an escalating series of demarches, or official letters registering U.S. discontent.

“It’s kind of like a warning shot with a popgun,” Ms. Vaughan said. “These half-hearted efforts under Clinton’s tenure produced no results, except for small progress with India. The other countries pretty much ignored them, except for China, which strung us along for years promising to cooperate, but not ultimately doing anything.”

Under intense bipartisan pressure from Congress, the Obama administration in recent years has stepped up its diplomatic engagement with recalcitrant countries. Letters have been fired off and meetings scheduled with the worst offenders, including China and Cuba.

But no resolution is in sight, and Democrats and Republicans on Capitol Hill raised the possibility of withholding financial aid, in addition to halting visas.

For now, however, the problem still remains.

“It’s very frustrating. Casey should still be here,” said Wendy Hartling, the mother of the Connecticut woman slain by Jacques.

She said she chiefly faults ICE, which she said should have done more to find ways to kick Jacques out of the country. A Homeland Security inspector general’s report earlier this year agreed, saying there were steps ICE didn’t take.

“I put the blame on ICE. That’s their job. Their job is to deport criminal aliens,” Ms. Hartling said.

The inspector general’s report did say that while agents at ICE’s Enforcement and Removal Operations (ERO) branch should have done more, including asking for the State Department’s help, they didn’t think they’d get anywhere with the diplomats.

“ERO officials believed that the Department of State would not intervene to encourage a foreign country to accept a violent offender like Jacques,” the inspector general said. “Although we did not interview State Department officials about this, we have no basis to believe that ERO’s experience in this area was unfounded.”

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