Soak the tombstone with natural spring water. Sulfites from a well, or chemicals from tap water, can cause a reaction to the calcite in the limestone.
Spray a non-ionic detergent onto the tombstone. A base liquid dish washing detergent should work, but read the label first. Non-ionic detergent has no ionic groups in its molecular makeup. This prevents the detergent from reacting poorly with hard water ions.
Sweep the tombstone with a natural or nylon soft-bristled brush. Use a mild amount of pressure while cleaning. Do not try to remove stains from the tombstone.
Brush detailed areas with a soft-bristled toothbrush.
Soak the tombstone several times with water to remove the detergent.
Repeat the process, if necessary. It is better to lightly clean the tombstone several times, than to give a hard cleaning once.