(On Friday night, after this article was published, Mr. Buttigieg’s campaign released more information about his McKinsey years, describing his projects in general terms without naming the clients. His assignments, it said, included work for a nonprofit health insurer in Michigan in 2007, a California environmental nonprofit in 2009 and, the following year, a logistics and shipping provider in Washington.

Speaking in Waterloo, Iowa, that evening, Mr. Buttigieg reiterated his request for McKinsey to release him from the nondisclosure agreement. “It’s not like I was the C.E.O. — I was making a lot of spreadsheets and PowerPoints — but people want to know from somebody who proposes to be the president of the United States, what’s in your past,” he said.)

Beyond Mr. Buttigieg’s agreement with McKinsey, this is something of an awkward moment to be associated with the consultancy, especially if you happen to be a Democratic politician in an election year shadowed by questions of corporate power and growing wealth inequality. The firm has long advocated business strategies like raising executive compensation, moving labor offshore and laying off workers to cut costs. And over the last couple of years, reporting in The New York Times and other publications has revealed episodes tarnishing McKinsey’s once-sterling reputation: its work advising Purdue Pharma on how to “turbocharge” opioid sales, its consulting for authoritarian governments in places like China and Saudi Arabia, and its role in a wide-ranging corruption scandal in South Africa. (All of these came after Mr. Buttigieg left the firm.)

Just this week, ProPublica, copublishing with The Times, revealed that McKinsey consultants had recommended in 2017 that Immigration and Customs Enforcement cut its spending on food for migrants and medical care for detainees.

After a campaign event on Wednesday in Birmingham, Ala., Mr. Buttigieg remarked on the latest revelations. “The decision to do what was reported yesterday in The Times is disgusting,” he said. “And as somebody who left the firm a decade ago, seeing what certain people in that firm have decided to do is extremely frustrating and extremely disappointing.”

On Thursday, a McKinsey spokesman said Mr. Buttigieg “worked with several different clients” during his time with the firm, but “beyond that, we have no comment on specific client work.”

But interviews with six people who were involved in projects that Mr. Buttigieg worked on at McKinsey, along with gleanings from his autobiography, fill in some of the blanks.