Depends on how soft the remaining mortar is. If you can scrape it out with your fingers, it would be better to remove it and replace it.
Assuming you do not want to paint, you might try an acrylic sealer used for tile and stone floors. It will leave a flat sheen (similar to varnish) but not cover up the look of the mortar.
I doubt a penetrating sealer (concrete waterproofer) would stop the deterioration of the mortar. but that would leave the brick unchanged in appearance if it did work.
Black dust is not the color I would expect, grey maybe but black??? Are you sure it is the mortar and not something else.
Also look into other grout sealing products.