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Perhaps no issue has bedeviled Republicans in recent election cycles more than illegal immigration. GOP candidates have been snared, over and over again, by the difficulty of appealing to conservative GOP primary voters who dominate the primary process while not alienating Latino voters, whose influence has grown in battleground states with each presidential cycle.

This week, as the controversy raged on about the anti-immigrant comments of Donald Trump, his fellow GOP contenders found themselves seared by the damage that he was inflicting on the Republican brand. When Trump first made the inflammatory assertion in his June announcement speech that some illegal immigrants from Mexico are criminals and "rapists," many of his rivals were slow to respond. But as the fallout has continued, more are speaking out -- trying to reinforce that Trump's views are not representative of the image of the Republican Party.

READ: Donald Trump digs in on immigration

Some, like Jeb Bush, have sharply criticized Trump's remarks, while others like Marco Rubio -- who has taken flak from conservatives for helping usher an immigration reform bill through the Senate -- have seemed unwilling to engage beyond paper statements. And all of them will be tested on the issue before a national audience during the upcoming debates that will weed out the weaker contenders from the field.

By infuriating many Latino voters, who have pointed to a number of studies showing that crime is actually lower among illegal immigrants than the broader population, the debate has brought unwelcome comparisons to recent struggles by Republicans to talk about immigration in a way that broadens their appeal among minority voters.

It revived talk, for example, of Mitt Romney's suggestion during the 2012 Republican primary that he was in favor of "self-deportation" for millions of illegal immigrants -- a statement that made it extremely difficult for him to draw large numbers of Latinos to support his campaign when it mattered later in the general election.

RELATED: GOPers run from Trump on immigration stance

President George W. Bush's strong performance among Latino voters -- he notched 44% of the Latino vote in 2004 compared to Romney's 27% in 2012, according to national exit polls -- seems like a distant memory to many Republicans now, despite a focus on building goodwill among minority voters by Republican National Committee operatives.

"We've gone from 44% to 27% among Hispanic voters, for a reason. You'll never convince me that it hasn't been about the way that we've handled this issue," South Carolina Republican Sen. Lindsey Graham said in a telephone interview with CNN on Tuesday.

"Mitt Romney showed a lot of political courage by saying that (his comment about) 'self-deportation' was a mistake, and now here we have Donald Trump casting 11 million people in a very derogatory manner. That's a problem," said Graham, who is also seeking the GOP presidential nomination. "There are some within the 11 million that are bad people, but I cannot tell you how harmful it is to reinforce a narrative that Republicans basically have very little respect for people."

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Immigration: The debate Republicans don't want ...