Republican lawmakers are starting to break with the Trump administration over its immigration crackdown in Minneapolis, as the party scrambles to respond to growing public anger after two US citizens were killed by federal agents.

But even as some Republicans in Washington speak out against the fatal shootings, they’ve avoided directly criticising President Donald Trump or his broader immigration agenda.

The emerging messaging on Minneapolis points to the party’s main dilemma heading into the midterm elections: whether and how to distance itself from the biggest controversies of Trump’s second term, while running on his overall record on immigration and other issues – a record that’s popular on the right, and that most Republicans helped push through Congress.

Their response to the fast-moving events in Minneapolis underscores the delicate balancing act.

So far, Republicans have focused criticism on US Immigration and Customs Enforcement, or ICE – the agency at the forefront of the Minneapolis operation – and other agencies within the Department of Homeland Security taking part.

But at the same time, Republicans have backed Trump’s immigration plans since his return to the White House, without highlighting their support.

Last year, the Republican-controlled Congress approved roughly $45 billion for border security and additional funding for interior enforcement to help the administration carry out Trump’s plan to deport millions of undocumented migrants from the country.

That support continued as opinion polls have consistently shown a majority of Americans believe ICE has gone too far in its tactics under Trump. A New York Times/Siena poll out last Friday found that 61 percent of voters believe ICE’s tactics have “gone too far.”

The survey came out two weeks after Renee Good, a US citizen, was fatally shot by a federal agent in Minneapolis, and one day before another American citizen, Alex Pretti, was killed by agents in the city, sparking nationwide protests.

In the face of public outcry over the killings, Senator Rand Paul of Kentucky, the chairman of the Senate Committee on Homeland Security & Governmental Affairs, called on Monday for the leaders of ICE and two other federal agencies to testify before his panel next month. The request came alongside calls from other Senate Republicans for more oversight of ICE after Pretti’s shooting.

“The tragedy and chaos the country is witnessing in Minneapolis is shocking,” Senator Lisa Murkowski of Alaska wrote on social media. “ICE agents do not have carte blanche in carrying out their duties.”

Murkowski’s condemnation does not come as a major surprise. The Alaska moderate is one of a small handful of Republicans in Congress who’ve been willing to openly criticise Trump since he first entered office. Paul also has clashed with the president and administration on issues in the past.

Senator Thom Tillis of North Carolina on Tuesday became one of the first prominent Republicans to call for DHS Secretary Kristi Noem to step down.

Noem “has proven to be way out of her depth,” Tillis told reporters, adding that her leadership in Minnesota “should be disqualifying.”

The White House has said Trump has confidence in Noem.

Moderate Republicans like Murkowski aren’t the only ones in the party criticising the immigration operation in Minnesota.

In recent days some of Trump’s allies in Congress also began speaking out against the shootings, though most issued carefully worded statements that didn’t explicitly mention the president.

“There are serious unanswered questions about federal use of force in Minnesota. Transparency and accountability are essential,” said Representative Max Miller of Ohio, a former Trump adviser.

Representative Andrew Garbarino of New York, the chairman of the House Committee on Homeland Security, called on the heads of ICE and other agencies to testify before Congress. Garbarino also came out in support of a full investigation into Pretti’s shooting.

“Congress has an important responsibility to ensure the safety of law enforcement and the people they serve and protect,” Garbarino said.

Senator Ted Cruz of Texas issued a more direct criticism of the administration after the Pretti shooting. But he focused on the administration’s rhetoric in response to the shooting and didn’t break with Trump’s larger agenda.

“What I think the administration could do better is the tone with which they’re describing this – that immediately when an incident like this happens, they come out guns blazing that we took out a violent terrorist, hooray,” Cruz said Monday on his podcast, Verdict with Ted Cruz.

The Republican response also is complicated by the gun ownership debate that has pitted the administration against the National Rifle Association, a traditional ally of the political right. A Trump-appointed federal prosecutor in Los Angeles said in a social media post after Pretti’s death that people who approach law enforcement officials armed should expect that there is a “high likelihood” they’ll get shot.

The NRA slammed the remarks. “Responsible public voices should be awaiting a full investigation, not making generalisations and demonising law-abiding citizens,” the NRA said in a statement.

Some Republican lawmakers defend immigration enforcement underway in Minneapolis and other US cities. Trump’s immigration policies remain popular with the party’s base, and after the recent shootings in Minneapolis some Republican voters in the state told the BBC they didn’t think the president should do anything differently.

“They’re just doing their job, which is right,” Jay Cielinski, said of ICE in an interview Tuesday in his hometown of Zimmerman, a roughly 50-minute drive north of Minneapolis.

The 59-year-old home inspector and self-described Trump supporter also said protesters, and state and local officials shouldn’t interfere with federal immigration operations, a view echoed by many Republicans including Trump and other senior administration officials.

“They should stay at home, stay out of the way,” Cielinski said of people who protest federal immigration operations. “Law enforcement officials at the city, state, county [levels] should be encouraging ICE to go in. They should expose the people that are illegal.”

Still, the cautious rebukes from Cruz and other prominent Republicans in Washington signaled an attempt to hold the administration accountable without alienating supporters like Cielinski.

The growing criticisms from Republican lawmakers came as Trump shifted his own tone in recent days after more facts emerged about the circumstances surrounding Pretti’s death.

After the shooting Trump initially called Pretti a “gunman,” and other senior US officials claimed the 37-year-old ICU nurse had turned up armed at an enforcement action to try and kill federal agents. But others have disputed that, saying Pretti was filming ICE agents with his phone then helping a woman pushed on the ground, and that he was not armed when he was shot.

Trump has since softened his rhetoric, and taken steps to try and ease tensions in Minneapolis. This week, he sent White House border tsar Tom Homan to take over the operation there, known as Operation Metro Surge. Homan’s arrival represented a shakeup in leadership, as top Border Patrol official Gregory Bovino is expected to leave the city.

Trump suggested on Tuesday that sending Homan to Minneapolis was intended to bring calm there. “We’re going to de-escalate a little bit,” Trump said in a Fox News interview.

The president also held calls on Monday with Minnesota Governor Tim Walz and Minneapolis Mayor Jacob Frey, both of whom are Democrats. Frey said afterwards that Trump told him the “present situation” in Minneapolis can’t continue.

White House officials pulled back their rhetoric, too.

“Nobody in the White House, including President Trump, wants to see people getting hurt or killed in America’s streets,” White House Press Secretary Karoline Leavitt told reporters Monday. “It is President Trump’s hope and wish and demand for the resistance and chaos to end today.”

The remarks appear to have opened the doors for more Republicans to voice concern about the shootings.

But the party’s future approach to its immigration messaging remains to be seen – both in the run-up to the midterms later this year and in the aftermath of the violence in Minneapolis.

Already, there are signs of a shutdown battle over immigration policy that could push lawmakers into their partisan camps.

Senate Democrats vowed to block a spending package if it includes additional funding for DHS, setting the stage for a possible government shutdown later this week. The fight could pressure Trump’s party to defend his immigration agenda and overshadow Republican calls for investigations into the shootings in Minneapolis.

Republicans also will likely continue taking their cues from Trump. How he responds will shape the debate in the days and weeks ahead.

Ana Faguy and Kristina Volk contributed reporting from Minneapolis.