Every four years, the AFL-CIO labor federation engages in a quixotic mission: to attempt to dissuade many of its own members from voting for their first choice for president, the Republican candidate. Between one-quarter and one-third pull the lever for the GOP, anyway.

The labor leaders' efforts have taken an added urgency this year because the Republican nominee, Donald Trump, may have the strongest appeal in decades of any GOP candidate to the union rank-and-file.

The AFL-CIO has conceded as much. An internal poll it did in June found that 41 percent of its members in five key battleground states — Ohio, Pennsylvania, Florida, Nevada and Wisconsin — favored Trump. More recently, it said that number had fallen to 36 percent following months of anti-Trump ads and messages it directed to members.

"The findings exposed another one of Donald Trump's lies — the idea that he has 'tremendous' support among union members," AFL-CIO President Richard Trumka said in a Sept. 20 speech. He added that organized labor had "unleashed the most comprehensive and sophisticated electoral program in our history" to educate its members about Trump.

More than a third of AFL-CIO members in those key states, according to its own survey, still intend to vote for the GOP candidate even after that campaign illustrated the gulf between the leadership of the nation's largest labor federation, which has never backed a Republican president in its history, and its members.

Matt Patterson, executive director of the conservative Center for Worker Freedom, said Democrats overall and the Clintons in particular have a history of support for trade and immigration policies that have hurt domestic manufacturing and driven down wages, key issues for unions. The GOP's support for similar policies has made that a non-issue in the past, but Trump has broken from that orthdoxy.

"Both in terms of policy and style, I think union leaders are terrified of Trump's appeal to their members," Patterson said.

Trump has a long, complicated relationship with unions, having dealt with them extensively as a casino and real estate developer. He's also a union man himself, having joined the Screen Actors Guild during his numerous appearances in TV and movies.

He has praised unions in the past. In his 2000 book The America We Deserve, he said, "Is Trump a union man? Let me tell you this: Unions still have a place in American society. In fact, with the globalization craze in full heat, unions are about the only force reminding us to remember the American family."

Trump also has tried to outflank Clinton on union-friendly economic issues. It is a weakness for Clinton, who once served on the board of non-union Walmart and whose husband signed the North American Free Trade Agreement in 1993, which many unions blame for job losses.

Trump has vowed to "rip up" NAFTA and denounced more recent trade deals such as President Obama's proposed Trans-Pacific Partnership, calling them job-killers.

"According to the Bureau of Labor Statistics, before NAFTA went into effect, there were 285,000 auto workers in Michigan. Today, that number is only 160,000. Detroit is still waiting for Hillary Clinton's apology," Trump said in his August economic policy speech.

Clinton has turned sharply against trade in this election, vowing to oppose the TPP, even though she helped to negotiate it as Obama's secretary of state. However, during the Democratic convention in Philadelphia, Virginia Gov. Terry McAuliffe, a close friend of the Clintons, said he expected she would back the trade deal after she got elected.

The Clinton campaign said he was "absolutely wrong," but the comment has forced her labor allies to do damage control.

Even Trump's anti-immigration stance isn't that far from where unions used to be on the issue. While the AFL-CIO now supports pro-immigration policies, in the 1970s and '80s, it viewed immigrants as unfair competition for domestic workers.

Even now, many of its constituent unions in the building trades and high-tech fields remain wary of policy proposals that would let in more immigrant workers. Service worker unions, on the other hand, are heavily represented by immigrants.

It is not clear exactly how many union members are Republicans, or at least vote that way, but it is likely that Trump's 36 percent in the AFL-CIO's five-state survey is above average for Republicans historically.

Presidential election polls rarely ask if a voter is a union member, preferring instead to ask, if they ask, if they are part of a "union household." That means that someone in the house is a union member but can also include spouses and other voting-age people who aren't.

Exit polls show that among union households, President Obama beat Republican candidates Mitt Romney by 58-40 percent in 2012 and John McCain by 64-36 percent in 2008. A 2004 CNN exit poll that did ask specifically about union members found that Democratic candidate John Kerry beat President George W. Bush 68-31 percent in that group.

The AFL-CIO is one of the few organizations that is closely monitoring union member preferences, but it is keeping a tight hold on its findings. It didn't announce the June survey results that Trump was getting 41 percent in the five battleground states at the time, revealing it only later in September when it said his support had slipped to 36 percent.

In a Sept. 20 email to reporters, Trumka announced that Trump's support among union members in Ohio was 32 percent, having fallen 12 points from June. The announcement didn't say how many were backing Clinton, however, and what the numbers were in the other four states. Spokesman Josh Goldstein declined to provide any more data.

Given that Trump's Ohio numbers were below the survey's average, that would suggest that he was doing better with union members in some of the other battleground states.

Trump's Ohio numbers may be rising. A late August poll by Suffolk University found that only 28 percent of Ohio union households were backing him.

Labor leaders have sought to counter Trump's appeal by pointing to past statements and actions by the candidate. The federation has circulated a picture reportedly showing Trump crossing a picket line by the International Alliance of Theatrical Stage Employees in 2004. The union's members had worked on his reality TV show "The Apprentice."

"The day you crossed our picket line was the day you showed me ... how little you care about American workers," Dan Mahoney, one of the strikers, said in an open letter the AFL-CIO circulated in September.

Service worker union Unite Here has held regular protests to highlight the case of the Trump Hotel Las Vegas, which is resisting an effort to unionize its staff. Bethany Khan, spokeswoman for Unite Here's Culinary Workers Union Local 226, said their outreach to immigrants helped to spark a 65 percent increase in citizenship applications in Nevada, a key battleground state.

"That's something the union worked really hard to do," she said.