WASHINGTON— Donald Trump promised to bring transformational change in a second term. The Senate is showing that there are limits to how much change he, and his supporters, can expect. The president-elect’s choice for attorney general, former Rep. Matt Gaetz , removed his name from consideration Thursday amid broadening Senate opposition within Trump’s own party. Meanwhile, his nomination of Pete Hegseth, the former Fox News personality, to be defense secretary is on the ropes.

Other cabinet picks, such as former Democratic Rep. Tulsi Gabbard to lead the nation’s spy agencies, are drawing scrutiny for their nontraditional views—in Gabbard’s case, for a perception that she has embraced the ideas of Russian President Vladimir Putin.

And some Republican senators have balked at Trump’s demand that they suspend public hearings and confirmation votes on some nominees and allow Trump to place them directly into office, using what’s known as recess appointments .

The botched Gaetz nomination, together with the other pushback, show that while GOP lawmakers generally agree with the president-elect’s ambition to reshape the federal bureaucracy, they are at times willing to buck their party’s standard-bearer.

“The president has the right to make the nominations that he sees fit, but the Senate also has a responsibility for advice and consent, and in this particular case, I think there was advice offered rather than consent,” South Dakota Republican Sen. Mike Rounds said of the failed Gaetz nomination.

A nominee needs a simple majority among the 100 senators to be confirmed. Vice President-elect JD Vance can break a tie. Republicans are expected to hold 53 Senate seats next year, meaning that a few defections can sink nominees.

Sexual-misconduct allegations snared Gaetz, who has been the subject of a long-running investigation by the House Ethics Committee into allegations that he had sex with a minor while he was in office, and have led to concerns about Hegseth, who was accused of sexual assault in 2017. Police in Monterey, Calif., on Wednesday released a report on that incident, saying that a woman sought an examination for sexual assault four days after an alleged meeting with Hegseth. Both Gaetz and Hegseth have denied wrongdoing, and charges weren’t filed in either case.

Trump’s pick for Health and Human Services secretary, Robert F. Kennedy Jr., has drawn praise for his ideas for improving nutrition and other aspects of public health but also criticism for his skepticism of vaccines, among other views.

Some Republican senators are voicing private concerns about Gabbard, who would be Trump’s director of national intelligence, according to one person familiar with the matter, raising the prospect of a new round of scrutiny for Trump’s picks.

When asked about Gabbard, Sen. Thom Tillis (R., N.C.) said the director of national intelligence should be someone who supports Ukraine in its defense against Russia, as he does.

“I’m going to have to have a very compelling story for anybody who’s going to influence policy in Ukraine,” Tillis said. “At DNI, I don’t know if that’s her or not, but when I get into the nomination process, I have no intention of supporting anybody who equivocates on support for Ukraine.”

Senate Armed Services Committee Chairman Jack Reed (D., R.I.) said that since Gaetz has withdrawn from consideration, scrutiny will “shift to these national-security picks, and that will, I think, raise additional questions about fitness.”

The early pushback from Senate Republicans could portend problems for Trump’s stated goal of overtaking other powers of Congress and the prerogatives of government institutions outside the executive branch, such as his expected moves to impound, or decline to spend, money appropriated by lawmakers for programs the president opposes, and to redirect funds elsewhere.

Tens of millions of Americans who voted for Trump were drawn by his criticism that the government had failed on several fronts and his promises of an overhaul.

When AP VoteCast, a large survey of people who voted in the presidential election, asked how much change voters wanted to see in how the country is run, some 56% of Trump voters said they wanted “substantial change.” An additional 37% said they would go even further, saying they wanted “complete and total upheaval.” That left a mere 6% saying they wanted small change or none at all.

For some, the personal qualities of some of Trump’s nominees, such as the allegations against Gaetz and Hegseth, have been less important than their promise to disrupt the way government has traditionally operated.

Bethany Beck, a Trump voter and retired mergers-and-acquisitions lawyer in Williamsburg, Va., thinks the Senate should give deference to Trump on the nominees and should discount the accusations against Gaetz and Hegseth, which she doesn’t trust as factual. She thinks Trump is right to reach for people with nontraditional résumés, such as Hegseth and Kennedy, to bring new thinking to the government.

“It’s clear from the last four years that there’s a considerable amount of structure in the government that doesn’t function. In health and safety, it failed us. In the economy, it failed us. In international affairs, it failed us,” said Beck, 76, referring to the government’s responses to Covid, inflation and wars in Ukraine and the Middle East. “And so the president is trying to rethink those things.”

By contrast, Gene Curran, a Trump voter outside Jacksonville, Fla., said he thinks the Senate should bring a robust scrutiny to Trump’s nominees. He said he opposed the idea of the Senate setting aside its confirmation powers to allow Trump to put people in office without scrutiny, through recess appointments.

“Any Republican who uses that as a strategy, I will not be voting for,” said Curran, 52, who owns an electrical contracting business. “That’s what I hired my senators to do—advise and consent.” He said the Senate should make sure Hegseth is qualified to lead the Defense Department and probe Gabbard for “giving Putin a pass on invading Ukraine.”

Curran thought Gaetz was “too morally challenged” to serve as the nation’s top law-enforcement officer, and he saw the failure of his nomination as proof that the advice-and-consent process works.

“It just demonstrates that our Founding Fathers were brilliant,” he said. “It gives me chills to think about it, so I can go on with my life and not worry about this whole Gaetz thing. It makes me proud to be an American. The system is functioning.”

Write to Aaron Zitner at aaron.zitner@wsj.com and Natalie Andrews at natalie.andrews@wsj.com