Yes, they usually do have a relationship to the tune being played. While improvising, the harmonic progression of the tune being played may cause the player to think of a melody, or fragment of a melody of a different tune which has the same (or similar) chord progression. Sometimes it is done to add a bit of humor to the proceedings. Maybe as an “inside” joke for the sake of the other players, or for a variety of other intents. For instance, in your clip Stormy Monday Blues, Muddy Waters sings “They call it Stormy Monday………Saturday I go out to play” and Clark answers with “It ain’t necessarily so, it ain’t necessarily so”. Great improvisers have very quick and creative minds, as you know.
The place where Wynton’s and the boys play is a very nice part of town. Lincoln Center is great. Other parts of the city still haven’t recovered from the COVID shutdown and things like public transportation have taken a big hit. Still impressive, no doubt.
Speaking of Clark Terry, Wynton’s boys and the plunger mute: