Displacement map is the way. Lets say you have 2 layers. Fog & image
Apply displacement map to your “image” layer, use the fog layer as a disp map. tweek the values, remember you may want to set the blending mode within the filter to luminance.
duplicate you fog layer and put it ontop of the image layer. select your image layer and set it to luma matte the fog layer.
This is a very simple techniq, but I