DMT recommends you use DMT products dry or with water only. Although mineral oil or WD40 will not harm the DMT products, they can impede sharpening if used in excess and can damage the sharpener overtime if not properly cleaned/removed from the sharpener.
DMT recommends that you always wipe your Diamond Sharpeners clean and store dry after use. Put away your benchstones dry and store in a dry area. Use leafing (paper towels, cloth, plastic storage bag) in between your bench stones to prevent abrasive sides from rubbing.