How do I stabilize brightness changes in Premiere Pro CC or Adobe AE?

One way:

Split the clip on the timeline in half with the blade tool at the mid point of the lighting transition.

Go to the last frame of the second half of the clip and use the Lumetri controls (Exposure, Contrast, Highlights,Shadow) to match the  the first half of the clip.

When done with the color correcting of the last frame of the second last half of the clip, return to the cut point between the two halves and place a  dissolve there. Adjust the duration of the dissolve to best cover the change in lighting.

Depending on the horsepower of your computer, you will likely have to render the clip to see your work playback in real time.

MtD

You would need to keyframe changes in the settings of Lumetri ... manually. There's no automated system for this.

Some people start something like this by doing the best correction for the whole piece on an "average" frame, then working forward/backward from that spot. I tend to do this starting from the beginning of the clip, correcting as I want it, then as it changes, using the Effects Control panel to keyframe the Lumetri effect ... making changes ... move forward, drop another keyframe ... it's tedious.

Also, for non-sudden shifts, you can do a 'cut' in the middle of the change (Ctrl/Cmd/k works fine) and then drop a cross-dissolve, pulling the ends to where the change stops. Correct one side, then the other, and the cross-dissolve handles the changing section.

Neil