Murphy's Oil Soap will remove grime, gunk, and old flaking varnish. Scrub using a washcloth over and over until the cloth no longer picks up the black stuff. Use a toothbrush to scrub out the checkering. Murphy's Oil Soap will remove varnish.
Another option for cleaning years of gunk from a stock is Meguiar's Fine Cut Cleaner
https://meguiarsdirect.com/products...kDHNbBLApvW_x8wdCcZU9alfapoDTPeHk6HZRHiL Howard Restore-A-Finish Dark Walnut does a remarkable job in blending out scratches and restoring faded finish.
Damascus barrels require no special care.
