How do I color correct footage shot with continuously changing white balance?

I have footage shot with a camera that changes its white balance as it shoots, so it switches between warm and cool. I have hours of it that I need to fix before I can key it (shot with an off-blue background, but sufficiently keyable) to put in a background, and I can’t key it if the color’s constantly changing. I can use either Premiere Pro or After Effects, how do I do it?

via Video Editing Forums: Digital Director – Adobe Premiere, Premiere Elements, and After Effects