USAFRet :
Generally, that would be done with a green screen.
Shoot the video, then just blot out the 'green'. Overlay that on the other video.
Taking moving parts out of one video otherwise is a frame by frame issue.
Agreed.
There MAY be software that can do this but I'm not aware of it.
For pictures I use the "Lasso" tool.
I tried to google "video lasso" and nothing showed up. Probably nearly impossible since the background would change so merging would be unlikely without gaps.
That's why they use a blue or greenscreen so they can merge together as the screen is a common color so it's easy to extract only the video objects that are a different color, ignore the screen and simply place the video object you want kept over any other video.