I think I've figured it out. I took a shot with a particularly bad instance of color-shifting problems and processed the DNG files with all of the settings sliders set to neutral in Camera RAW and there was no color-shifting. So, I think the root of the problem is how the settings sliders in Camera RAW (like Clarity, Highlights and Shadows) dynamically adjust how they affect an image based on the content of the image. This isn't a problem when it comes to editing still images because the content of the image isn't changing - but with video, it is. I'm going to experiment a little more and see if it's maybe just one of the settings sliders that acts this way...