It is a method of transitioning you into the next scene. Anne Coates, the editor, suggested the idea to David Lean on "Lawrence of Arabia" an idea of preceding or overlapping audio ahead of the picture cut. She had apparently seen this idea used in numerous French films. Lean liked the idea and used it several times in the film to great effect. One example was the scene at the Suez Canal when the motorcyclist shouts "Who are you?" (which I heard was Lean's voice!). We then cut to Lawrence' face and an audio effect of a ringing bell is on the soundtrack. Due to the timing, before the audience has had a chance to analyze what has happened we cut to a busy noisy street. "Lawrence" may not claim to be the first, but is the first film that I recall experiencing it and a variation on it, including the use of dialog from the following scene, has been used by many directors and editors since as a way of hiding the effect of picture cuts unless they want the audience to notice it for shock value. It was originally described as a 'French cut'.
This may not answer your question, but I feel that the origins of what you're describing started here.