Non-ionic detergents such as D/2 Biological Solution, NP40, Triton X-100, Orvus, or Tween20 are recommended by many experts for cleaning gravestones. These chemicals are electrically neutral cleaning agents that neither contain or contribute to the formation of soluble salts. They are neither soapy (normal soaps are Ionic detergents), nor do they affect pH. By decreasing the wash water's surface tension non-ionic detergents reduce droplet formation on the stones surface. In general they have been proven to provide a better overall wetting of the stones surface that other detergents and as a result, produce better results in the removal of general soiling.

Non-ionic detergents are available from conservation, janitorial, and photographic suppliers in various sizes.  You may want to bring a small spray bottle of water for gently cleaning dirt and debris from the stone. Wash stone with solution, then rinse stone with clean water.