You can put a music track on Screencast-O-Matic. Their blog explains how.
You do not have a lot of control over the clip. It appears you can only (easily) get one clip and can just adjust the volume and whether it loops. You can have a fade in or fade out of a few seconds. In one video I did get two different clips in. I think it was a mistake that had to do with a cut I made later. At any rate I couldn’t replicate a music change later. So, if it is a feature it is not intuitive.
You can have narration and music, but it makes kind of an audio mess. I only plan to use this right now when I chose not to narrate as a signal to students that there will be no narration. Then they do not spend time worrying about their audio settings.
Here are three videos I made last night playing with this feature. The one about the ladder is the one where somehow I got two different songs. The transition point is a little ugly though.