D5200 Color levels change through video
Don't use auto white balance and use manual exposure.

Auto white balance will guess at what is white and may shift while you are running video on some cameras (which makes sense since often people may move from one room to another while shooting or lighting conditions could change.

Manual exposure will preserve luminosity.  Auto exposure shouldn't change color temperature (which is what you are noticing) however it will change how bright things are and that can be just as much of an issue for trying to get accurate green screen footage.