“People are realizing we are not as exceptional as we thought,” Ms. Hume said of the United States.

Democracy rankers have taken note. The Fund for Peace, a nonprofit that focuses on fragile states, declared the United States the fourth-most-worsened country for 2018, after Qatar, Spain and Venezuela. In 2017, the Economist Intelligence Unit, the research division of The Economist Group, downgraded the United States to a “flawed democracy,” from a “full democracy,” citing declining popular trust in government that began long before Mr. Trump’s election.

Daniel Noah Moses, director of educator programs for Seeds of Peace, a nonprofit that began with work on the Middle East, but recently has ramped up its focus on the United States, said when he moved to America from Jerusalem in 2017, the political climate seemed strangely familiar.

“I’ve been surprised by how similar it all is — the gaps in understanding, the levels of emotion, the negation of ‘the other,’” he said.

Dr. Green’s homegrown peace mission consisted of 18 people from Leverett, a liberal enclave in western Massachusetts, and 11 people from eastern Kentucky. In three-day sessions in both places, Dr. Green used tools from social psychology to probe underneath politics. The goal was not to change minds, but to broaden them, by getting the participants to see one another as people.

The beginning was bumpy. The initial overture for a meeting with the Kentucky residents came from Jay Frost, a retired corporate training consultant in Leverett who admits that he did not think much of Trump voters at first.

“‘Stupid’ was the adjective I used,” he said, explaining his early thinking.

He wrote in an email that he wanted to understand “how rural white voters could possibly support such a vulgar, dystopian presidential candidate,” language he says he now regrets.