There are many types and styles of live caricature. One that I've tried recently is the "hit and run" caricature. In this case, instead of being front and center drawing at an easel and putting on a show, the artist is to remain as invisible as possible, sketching subjects without their knowledge and surprising them with a drawing of them as a result.
There's a real trick to doing it. You have to be far enough away so that the subject can't see you and draw fast enough so that they don't walk away in the middle of the picture and you can't find them. Facial likeness becomes secondary to body pose, posture and gesture.
I try coming up with stories for mine, give them a bit of a narrative: