When you say “shifting”, do you mean the colors are interacting oddly? or the position of the grad color “nodes” is changing position relative to the shape layer?
If the former, make sure your comp is a high enough bit depth so that when the subtle variations combine with other layers’ grads there is in fact a color they can map to in that larger color space, otherwise banding can occur.
If the latter, are you sure you are moving the whole layer and not just the mask? (assuming some of the layers might be solids and not shape layers) With all the layers you have going you might not notice that you had . The “shape” would move but the grad would stay where it was, because the layer itself isn’t moving. If they are all true shape layers I don’t think this would be the problem.
Also check the “align with layer” button in the gradient parameters. With it “off” the layer can move independent of the gradient’s position.