How to create a video recording of a background window I'm looking for some solution for creating a video recording limited to a single window, which continues to work even if that window gets partially or fully covered by other windows.

All of the screencasting/recording solutions I found seem to, at most, allow limiting the recording by a sub-region of the screen, and can't record covered/hidden windows.
No applications yet exist for recording movies from the content of obscured windows.

Screen recorders like QuickTime, ScreenFlow, and Snapz Pro X grab from the graphics hardware after composition. The process is focused on speed and low impact on the operating system.

On OS X, processes only get limited access to window content from other processes.

Hidden or obscured windows can be captured but doing so is relatively slow, computationally expensive, and not suitable for recording moving content. This is why tools that do offer individual window capture limit themselves to grabbing still images.