ATLANTA — Commemorative events for the Rev. Martin Luther King Jr. slid seamlessly into celebrations of the swearing-in Monday of the nation’s first black president, with many Americans moved by the reminder of how far the country has come since the 1960s.

‘‘This is the dream that Dr. King talked about in his speech. We see history in the making,’’ said Joyce Oliver, who observed King Day by visiting the National Civil Rights Museum in Memphis, built on the site of the old Lorraine Motel, where King was assassinated in 1968.