Donald Trump has jumped out in front of the 2016 Republican primary race largely on the strength of his anti-immigrant rhetoric.

Now, in the quest for an edge in a tightly packed field, some Republican candidates may be drawing nearer to exploiting an even older fault line in American society.

With increasing frequency over the last week, Republican candidates have sallied attacks on Black Lives Matter, the movement born of protests against police killings of African Americans. The Republican candidates have been testing new messages with various subtle edges – and abilities to cut.

Republicans have stopped short, so far, of tying Black Lives Matter by name to the killing of multiple law enforcement officers across the country last month. But at least two Republicans, Scott Walker and Ted Cruz, have cited the group as evidence of a new anti-police “culture” – created by President Barack Obama, they say – that has endangered the lives of law enforcement officers.

Other Republican candidates have faulted Black Lives Matter as selfish in its focus on African Americans, as inflammatory in its rhetoric or as failing to focus on black-on-black violence. Drowning them out and goading them on have been Fox News, Rush Limbaugh and other conservative barkers, who have not hesitated to draw an explicit link between the movement and the killing of police officers, despite a lack of any supporting evidence.

“That kind of incident is seized upon by Black Lives Matter and other radical groups to demonize the police,” Fox News host Bill O’Reilly said this week, of a fatal shooting by police of a Latino man. “Every time there is a controversy about an officer shooting a black person, they’re out there stirring the pot.”

O’Reilly’s guest pointed out that fatal attacks on law enforcement are down 17% so far in 2015 when compared with last year. But the facts have not stopped Fox News from branding Black Lives Matter a “murder movement” and a “hate group” – the latter label having first been applied by Limbaugh.

Only a small minority of the current Republican field have tied the activists to violence. Some have offered rebranding advice – “I think they should change their name maybe – if they were All Lives Matter, or Innocent Lives Matter,” Kentucky senator Rand Paul said last week. While others have offered a historical perspective; commenting on the network’s name, former Arkansas governor Mike Huckabee has said he thinks Martin Luther King Jr would “be appalled by the notion that we’re elevating some lives above others”.

But the sharper rhetoric is disturbingly familiar to some experts, who see the candidates and their media counterparts as playing on race in ways discouragingly reminiscent of some of the less distinguished episodes of national presidential politics.

Fredrick C Harris, director of the center on African American politics and society at Columbia University, said that Republican candidates might be seen as taking a page from Richard Nixon’s 1968 presidential campaign to mobilize white voter resentment – with a dual appeal to “law and order” and to a “silent majority” mistrustful of black empowerment.

“Just as the success of the civil rights movement fueled white racial resentment on the right, the Obama presidency coupled with the BLM movement is fueling a backlash,” Harris said in an email. “However, the demographics in the overall electorate have changed in a way that could hurt the Republican nominee in the 2016 general election.

“But the candidate who bashes Obama and BLM may win in the short term by building support among Republican primary voters who resent the BLM movement and its criticism of police.”

Whether playing to an audience already in place or building a new one, conservative media noticeably accelerated its (negative) coverage of the Black Lives Matter movement following the fatal shooting last weekend of a deputy sheriff in Harris County, Texas.

Fox News interwove its coverage of the shooting with video footage of a Black Lives Matter march Saturday outside the Minnesota state fair – despite local Texas authorities saying there was no known link between activists and the killing.

In the footage, marchers chant against the police: “Pigs in a blanket! Fry ’em like bacon!” The governor condemned it as “a terrible thing to say”. A local BLM organizer later said the marchers had been joking all day with police escorting them and “it was more playful than anything”.

But the context was lost in endless replays on television, and later in a statement from Wisconsin governor Scott Walker, a Republican candidate. In an editorial on the website Hot Air published Wednesday, Walker diagnosed “a disturbing trend of police officers being murdered on the job” and blamed “a rise in anti-police rhetoric” “under President Barack Obama”.

“This kind of attitude has created a culture in which we all too often see demonstrations and chants where people describe police as ‘pigs’ and call for them to be ‘fried like bacon’,” Walker wrote. “This inflammatory and disgusting rhetoric has real consequences for the safety of officers who put their lives on the line for us and hampers their ability to serve the communities that need their help.”

On Saturday, Texas senator Ted Cruz, also a presidential candidate, seized on the chant in a weekend radio interview.

“Much of this is the fruit of the divisiveness and vilification of law enforcement that we’ve seen throughout the Obama administration,” Cruz said. He said that police officers were reluctant to do their jobs out of fear of criticism.

The latter assertion was amplified on Wednesday by South Carolina governor Nikki Haley, who is not running for president. “Most of the people, who now live in terror because local police are too intimidated to do their jobs, are black,” Haley said in a Washington speech. “Black lives do matter, and they have been disgracefully jeopardized by the movement that has laid waste to Ferguson and Baltimore.”

In the wake of the incident, Deray McKesson, a BLM activist, condemned “prejudicial statements about motive that are baseless”.

Not all criticism of the movement from the Republican side has amounted to fearmongering. Ben Carson, a neurosurgeon, a Republican and the sole African American in the presidential race, said in an appearance in Harlem last month that Black Lives Matter amounted to finger-pointing and a distraction from real problems.

“Of course black lives matter,” Carson told local politicians and businessmen. “But what I feel instead of people pointing fingers at each other and just creating strife, what we need to be talking about is how do we solve problems in the black community.

“Of murder, essentially.”

Additional reporting by Nicky Woolf