So how does it work? Do they correct scene per scene? I.e., they pick a frame from a scene, fiddle with it, and then batch-process the whole scene based on that one frame?
Would you say that colour-correction software is more or less powerful than its still-images counterparts?
Depends on what you mean by powerful. They both do fundamentally the same thing, even if they may use different terms for the same adjustment.
The main difference with motion grading is the ability to key areas, such as a face or a window that will then be affected by whatever treatment is applied as the camera or subject moves. The software can smartly keep the adjustment to just the defined area even as it changes shape/size. This saves having to mask the area in each frame individually.
You can use
DaVinciResolve for free if you like, very well regarded software for motion work. They only charge for the
control hardware to run it, which any serious user will buy.
What is also very important is that different shots from the same scene match. Particularly if using different cameras in say an action scene with disposable 5Ds and GoPro being mixed with Alexa footage.
Even different lenses on same sensor can give different results, particularly if different brands. This is why you tend to use a set of cine lenses that have t-stops and not f-stops and the same colour rendition, as well as same sizing to allow quick swapping between takes without re-rigging camera.
Re F + T stops, think o f them as fictional and truthful. An f-stop is the theoretical transmission of light, the T-stop is the actual transmission of light. My f2.8 24mm prime is much brighter than my f2.8 16-35mm L lens for example so if shooting a scene with both set to f4, then the exposures will be quite different.