When Sean Hannity of Fox News appeared onstage at a rally with President Trump — and called the press corps “fake news” from the podium — it was the culmination of the network’s shift from its “fair and balanced” founding days to a post-Ailes MAGA messaging machine.

When Sean Hannity of Fox News appeared onstage at a rally with President Trump — and called the press corps “fake news” from the podium — it was the culmination of the network’s shift from its “fair and balanced” founding days to a post-Ailes MAGA messaging machine. By Jim Rutenberg, Robin Stein, Margaret Cheatham Williams, Whitney Hurst and Jonah M. Kessel

Rupert Murdoch, the founder of a global media empire that includes Fox News, has said he “never asked a prime minister for anything.”

But that empire has given him influence over world affairs in a way few private citizens ever have, granting the Murdoch family enormous sway over not just the United States, but English-speaking countries around the world.

A six-month investigation by The New York Times covering three continents and including more than 150 interviews has described how Mr. Murdoch and his feuding sons turned their media outlets into right-wing political influence machines that have destabilized democracy in North America, Europe and Australia.

Here are some key takeaways from The Times’s investigation into the Murdoch family and its role in the illiberal, right-wing political wave sweeping the globe.

The Murdoch family sits at the center of global upheaval.

Fox News has long exerted a gravitational pull on the Republican Party in the United States, where it most recently amplified the nativist revolt that has fueled the rise of the far right and the election of President Trump.

Mr. Murdoch’s newspaper The Sun spent years demonizing the European Union to its readers in Britain, where it helped lead the Brexit campaign that persuaded a slim majority of voters in a 2016 referendum to endorse pulling out of the bloc. Political havoc has reigned in Britain ever since.

[Read the complete investigation into the Murdoch family.]

And in Australia, where his hold over the media is most extensive, Mr. Murdoch’s outlets pushed for the repeal of the country’s carbon tax and helped topple a series of prime ministers whose agenda he disliked, including Malcolm Turnbull last year.

At the center of this upheaval sits the Murdoch family, a clan whose dysfunction has both shaped and mirrored the global tumult of recent years.

The Times explored those family dynamics and their impact on the Murdoch empire, which is on the cusp of succession as its 88-year-old patriarch prepares to hand power to the son whose politics most resemble his own: Lachlan Murdoch.

A key step in that succession has paradoxically been the partial dismemberment of the empire, which significantly shrunk last month when Mr. Murdoch sold one of his companies, the film studio 21st Century Fox, to the Walt Disney Company for $71.3 billion.

The deal turned Mr. Murdoch’s children into billionaires and left Lachlan in control of a powerful political weapon: a streamlined company, the Fox Corporation, whose most potent asset is Fox News.

Mr. Murdoch nearly died last year, making the succession question an urgent one.

Succession has been a source of tension in the Murdoch family for years, particularly between Mr. Murdoch’s sons Lachlan and James.

His two sons are very different people. James wanted the company to become more digitally focused and more politically moderate, while Lachlan wanted to lean into the reactionary politics of the moment.

[Read the complete investigation into the Murdoch family.]

The brothers have spent their lives competing to succeed their father, and both men felt as if they had earned the top job. When Mr. Murdoch decided to promote Lachlan over James, it was Lachlan who delivered the news to James over lunch, souring the already poor relationship between the men.

James briefly quit the company in protest. But he was lured back by a carefully crafted compromise that put Lachlan in charge but allowed James to save face by maintaining the public illusion that he was the heir.

But all of these succession plans — as well as the lucrative Disney deal — were thrown into chaos last year when Mr. Murdoch broke his spine and collapsed on a yacht.

He was rushed to a hospital, and appeared to be so close to death that his wife, the model Jerry Hall, summoned his children to say their goodbyes.

Mr. Murdoch survived, but his brush with death only highlighted the instability in his family — and at the heart of his empire.

The Murdoch empire has been a cheerleader for the American president and helped overthrow an Australian prime minister.

Mr. Murdoch’s media outlets have promoted right-wing politics and stoked reactionary populism across the globe in recent years.

During the 2016 campaign, the Fox News host Sean Hannity advised the president’s former lawyer, Michael D. Cohen, to be on the lookout for ex-girlfriends or former employees of Mr. Trump lest they cause him trouble, according to two people who know about the interactions (Hannity denies offering such advice). Mr. Cohen was later sentenced to three years in prison for paying hush money to two women who said they had affairs with Mr. Trump.

[Read the complete investigation into the Murdoch family.]

The Murdoch empire has also boldly flexed its muscles in Australia, which was for many years Lachlan’s domain.

In Australia, Lachlan expressed disdain for efforts to fight climate change and once rebuked the staff at one of his family’s newspapers, The Australian, for an editorial in support of same-sex marriage (He says through a representative that he is in favor of same-sex marriage). He also became close to the politician Tony Abbott, whose 2013 election as prime minister was given an assist by Murdoch newspapers.

The Murdoch family changed Australian politics in 2016 when it took control of Sky News Australia and imported the Fox News model. They quickly introduced a slate of right-wing opinion shows that often focused on race, immigration and climate change. The programming became known as Sky After Dark.

Last year, Mr. Turnbull and his staff accused Rupert and Lachlan Murdoch of using their media outlets to help foment the intraparty coup that thrust him from office in August. Mr. Turnbull, a moderate and longtime nemesis of his friend Mr. Abbott, was replaced by the right-wing nationalist Scott Morrison.

The Murdochs have denied any role in Mr. Turnbull’s downfall.

James Murdoch thought Fox News was toxic to the company.

James Murdoch became disillusioned with the family empire in the years before Lachlan emerged as heir. He came to see Fox News, in particular, as a source of damaging ideological baggage that was hobbling the company’s efforts to innovate and grow.

But Lachlan and Rupert did not share that belief. When Roger Ailes, the chief executive of Fox News, was ousted in 2016 amid a sexual harassment scandal, James wanted to revamp the network as a less partisan news outlet. He even floated the idea of hiring David Rhodes, a CBS executive.

[Read the complete investigation into the Murdoch family.]

His proposals went nowhere. Lachlan and Rupert opposed any change to what they saw as a winning formula and decided to stick with Fox’s incendiary programming.

But James believed he had seen firsthand the damage that outlets like Fox News were doing to the company.

He was the face of the Murdoch empire in Britain during a 2010 attempt to take over British Sky Broadcasting, in which the company owned a minority stake.

That bid was blown to pieces by the 2011 phone hacking scandal, which forced James and his father to appear before Parliament to explain why their employees hacked into the voice mail of private citizens, including a dead 13-year-old girl. The scandal forced the Murdochs to abandon their bid for Sky.

Five years later, facing pressure from digital rivals like Netflix and Amazon, the family made a second bid for Sky. James again acted as the empire’s public face. The bid again collapsed in a humiliating scandal.

This time it centered on the culture of Fox News, where sexual misconduct allegations and millions of dollars in secret settlements led to the departure of Mr. Ailes, the star host Bill O’Reilly and Bill Shine, an executive who later went to work for President Trump.

The behavior of Mr. Hannity, who used his show to spread conspiracy theories about the death of a Democratic National Committee staff member named Seth Rich, also fed concerns in Britain over the ethics of the company.

After months of review by regulators — and scrambling inside 21st Century Fox — the British government issued a withering rebuke to the Murdochs last year.

Not only did Britain block the company’s bid for Sky, it also ruled that no member of the Murdoch family could serve at Sky in any capacity, including on its board. At the time, James was serving as Sky’s chairman.

It was a deep humiliation that convinced James once and for all that the family empire could not survive its own politics and culture. Lachlan instead saw it as validation of his belief that James, having failed to acquire Sky once, had been the wrong man for the job.

Either way, by putting a much-needed revenue stream permanently outside the family’s grasp, it helped make the sale of 21st Century Fox inevitable.

The Disney deal worsened a family rift.

James and Lachlan were bitterly split over the prospect of selling 21st Century Fox to Disney. James pushed hard for the deal, which was completed last month, and Lachlan fiercely opposed it.

Lachlan vociferously opposed the deal because it substantially shrank the company he returned from Australia to one day lead, people closer to his brother said. He felt so strongly that at one point he warned his father he would stop speaking to him if he continued to pursue the deal. Mr. Murdoch ignored that threat. (Lachlan denied making the threat.)

[Read the complete investigation into the Murdoch family.]

Lachlan’s opposition was also fueled in part by his suspicion that his brother’s judgment had been clouded by personal ambition, people closer to Lachlan said. He thought James was willing to sell 21st Century Fox for less than it was worth because he wanted the deal to include a job for himself at Disney.

The deal transformed Disney into a media colossus, and a job there might have enabled James to position himself as a successor to its chief executive, Robert A. Iger. It would also let him escape the family company, its political baggage and the prospect of working for Lachlan.

The two brothers clashed over everything. When James wanted to respond to President Trump’s 2017 travel ban with a statement reassuring their company’s Muslim employees, Lachlan strenuously resisted. When James bought their father’s Beverly Hills mansion for $30 million, Lachlan, who had also wanted the house, got so upset that their father gave him some of its antique furniture. James thought he had bought that, too.

During the Disney negotiations, Mr. Murdoch grew concerned enough that James’s ambitions might interfere with the deal that he decided to assure Mr. Iger that it was not conditional upon Disney’s hiring his son. In the end, the sale went through, but James did not get a job. Today, the two brothers are barely on speaking terms.

Three of the Murdoch children wanted out, and Lachlan might too.

After the Disney deal, the commitment of Mr. Murdoch’s children to what remains of his media empire has been called into question.

The Disney deal made all of them an enormous amount of money: Mr. Murdoch received $4 billion and his children received $2 billion each. As executives at 21st Century Fox, Lachlan and James got an additional $20 million in Disney stock plus golden parachutes worth $70 million each.

[Read the complete investigation into the Murdoch family.]

Mr. Murdoch had structured his companies, 21st Century Fox and News Corporation, so that the Murdoch Family Trust held a controlling interest in them. He held half of the trust’s eight votes, and the remaining four were divided up among his four adult children. They were barred from selling those shares to outsiders.

James struck out on his own at the end of 2018. To make a more complete break with the company, he and his sisters Elisabeth and Prudence offered to sell their shares to Lachlan.

Mr. Murdoch embraced the idea and urged Lachlan to buy out his siblings. Then father and son would own the company together.

Bankers drew up documents to execute the sale, but Lachlan backed out; he said that it was not financially doable, though the decision raised questions about his commitment to the company.

People close to James said they believed Lachlan was not sure he wanted to stay at the company after the Disney deal was complete. They said he might even want to go back to Australia.