Video editing. This is not my day job so I want to spend little money on it. So here are some cheap options, especially free/libre video editors. New open-source ones are launched with curious regularity and get quite a lot of hype. I do not do enough video editing to have opinions upon the dynamics of this.
MLT is a video editing framework. There are at least two actual video editors built upon it, which both seem to be fairly actively developed.
Macos/Linux) also edits video.
OBS is not a video editor, but it might do what you want nonetheless; since it facilitates live composing, recording and streaming of multiple video and audio sources.
Routing video live
See video routers.