Vary thickness of non-vector lines

Hey all, working on a custom effect to turn images into pencil drawings. It's for an animation so it has to be procedural (I can't be custom applying techniques to thousands of frames).
Got it working pretty well so far, but there's one missing ingredient I can't work out:

Say I have linework (not vector or stroke) - how can I vary the thicknesses of the line?
Turbulent Displace doesn't work because that twists the line, and Minimax is not controllable enough (doesn't do decimals so it's either way too much, or off).
I can make it so we have alpha around lines if needed.
Doesn't need to be much either.

Any thoughts?

nourishingmedia.com

Thanks Frank, that actually does work fairly well! Just make it a thin border scaled large. I didn't think it would be even on each side of a line (which would just make it wobbly) but it is!

It's a pain in the butt to have to key out the white background each time, but it's better than nothing. (Currently, using Invert > VC Colour Vibrance with Matte Alpha > Invert, but really thought there'd be an easier way)
But definitely better than nothing - thanks Frank!

nourishingmedia.com