How to correct varying luminance levels in a video

If I understand correctly, this is a primary color correction problem concerning brightness alone. In addition, it should be dynamically treated with keyframes for each change that occurs. The classic steps in these cases is to study the waveform to understand the values that change over time. Then you enable the automatic keframes and compensate the brightness of the various timecodes through Color 3 Way (only the slider of the contrast). If you want to be more precise (but it's longer and more complicated) you can use Histogram-Bezier (master) or, if you want to be fast and approximate: Contrast/Brightness. The important thing is that the modification is done by checking on the waveform.

Thanks Andrea, the Color3Way does the trick. I presume you mean the slider called "values" . I selected "copy to all" of the middle one, then adjusted at a lot of points the value. Not perfect, but a huge improvement.

It so happens that in this video there is a piece of wall that could be used as an indication for how dark/light it should be, once you have established reference frame. I guess there is no way this can be automated by using the color picker or some such? I don't know if any other NLE has such a feature.

For "shot matching scene to scene", CinGG does not have any storyboard, automation or possibility to store grading presets as other programs do. It is not a serious lack because in any case you have to make manual adaptations and so much is worth doing everything in manual (often it is also faster and more accurate). See book by Alexis van Hurkman: Color Correction Handbook.
You can create LUT with the appropriate ffmpeg plugin (which I don't know how to use), or first create a Haldclut (which is easier) and then turn it into 3D LUT. In this second case you can see:
https://www.cinelerra-gg.org/forum/how-to/color-matching-scene-to-scene/