After a meeting with Donald Trump, Martin Luther King III said Monday that he believed the president-elect intended to represent all Americans but that champions of the country’s poor and marginalized communities must keep up the pressure on government.

“Certainly he said that he is going to represent Americans. He’s said that over and over again. We will continue to evaluate that,” Mr. King, the oldest son and oldest living child of civil rights leader the Rev. Martin Luther King Jr., told reporters at Trump Tower in New York.

“I believe that’s his intent,” Mr. King said of Mr. Trump. “I believe we have to consistently engage with pressure, public pressure. It doesn’t happen automatically. My father and his team understood that, did that.”

Mr. Trump met with Mr. King to discuss his father’s legacy on Martin Luther King Jr. Day.

Mr. Trump escorted Mr. King to the lobby, where they shook hands and said goodbyes before the president-elect got back on the elevator without talking to reporters.

Asked about the feud between Mr. Trump and civil rights hero Rep. John Lewis, Mr. King said that “in the heat of emotions, a lot of things get said on both sides.”

Mr. King, who has continued his father’s work as a human rights activist, said that the focus should be on improving the lives of impoverished Americans, which is what his father would have focused on.

“I think my father would be very concerned about the fact that we have 50 [million or] 60 million people living in poverty, and somehow [we’ve] got to create the climate for all boats to be lifted,” he said. “In America — a multitrillion-dollar economy — it’s insanity that we have poor people in this nation. That’s unacceptable.”

He said that he disagreed with Mr. Trump’s characterization of Mr. Lewis as “all talk, no action.”

“Absolutely I would say John Lewis has demonstrated that he is action,” said Mr. King. “At some point in this nation, we’ve got to move forward.”

Mr. Trump came under fire from Democrats over the weekend for Twitter posts critical of Mr. Lewis. The Georgia Democrat said Mr. Trump would be an “illegitimate president” and that he was joining a boycott of the inauguration.

The president-elect later canceled a visit scheduled for Monday to the National Museum of African American History and Culture. He cited scheduling conflicts, but Democratic critics said it was due to blowback from his clash with Mr. Lewis.

In a forceful response to Mr. Lewis’ remarks Saturday, Mr. Trump fired off a series of tweets saying the congressman should focus on problems in his Atlanta district. He tweeted: “All talk, talk, talk — no action or results. Sad!”

Mr. Trump also tweeted: “Congressman John Lewis should finally focus on the burning and crime infested inner-cities of the U.S. I can use all the help I can get!”

His characterization of Mr. Lewis, who led the 1965 “Bloody Sunday” march in Selma, Alabama, and his description of Atlanta prompted some Democrats, including prominent black lawmakers such as Sen. Cory A. Booker of New Jersey, to accuse Mr. Trump of being racially insensitive.

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