Simple rule of using anchors... if you use an anchor, you should never tween or animate anything above that anchor, nor the anchor itself. Only child objects. The reason is obvious and you've ran into it -- anchor always tries to reposition its game object. That's what the script is supposed to do, after all.
Disabling anchors would be one way. Another, better way, is to animate panels below your anchors separately. For example in Windward I have dialogs anchored to the center, left, and right, and all animate independently of one another. If I want to show more than one, I simply play more than one animation, moving them into view.