I think I understand the reason for the problem.
Your lens is not a constant-aperture lens. Thus, when you zoom to a longer focal length the aperture stays at the same diameter and would make a darker image. Normally, the camera controls the aperture when making an exposure so that a setting of f/8 would be controlled properly regardless of the chosen focal length. This is done each time an ordinary photo exposure is made, as the lens is wide open during the time prior to pressing the shutter release button.
My assumption is that, when making a video, the camera controls the aperture at the time that the video is started but does not correct for the changing f-stop value when zooming as the video is progressing.
To verify my theory, try the same test with another consumer-grade zoom lens and also try it with a constant-aperture zoom lens such as the 24-70 f/2.8L. I suspect you won't see the change with the 24-70 but will with any zoom lens that has two f-stop values shown on the front of the lens.