Just saw this so don't know if anybody is still looking.
I have a lot of experience cleaning stones as I volunteer at a historic cemetery and have been trained by a professional.
First off, you need to be the owner of the stone or otherwise have legal access (such as being a family member) and permission from the cemetery. If it's a private cemetery they can make any rules they want and you need to know them. Public cemeteries are usually covered by local law. If you do not own the stone you could find yourself in big trouble if you just go at it.
That said, what has been said about, generally, not using anything but water and a soft brush is a good general rule. You need to also examine the stone carefully to make sure it is stable on its base so you won't knock it over, does not have serious cracks, is not of a very soft stone that is chipping, and is not something delicate like slate. If it is stable, generally sound, and not slate or crumbling, you can proceed with care.
First off, you only want to work on a stone in the warm months and, if possible, when the sun is not directly shining on it. You will also need lots of water as you'll want to soak anything that is to be removed to soften it.
We use a soft rubber putty spreader, wooden popsickle sticks, and wood cuticle sticks to remove bulk lichen and moss. The cuticle sticks are good for getting into lettering. An old toothbrush is also good for getting into tight areas.
You should not use any type of cleaning solution, soap, shaving cream, anything abrasive, or pressure sprayers.
Pros (us included) use a product called D/2 which is especially formulated for cleaning stone and also kills moss and lichen. It's been used on many national monuments and the White House. You can buy it online. It's really expensive but a little goes a long way and we generally dilute it 50% with water. Follow the directions. Generally, you spray some on, wait five or ten minutes, use your soft brush to scrub, and then rinse with lots of water. You might have to do this several times if the stone is really dirty. If a little stays on that's OK as it will continue to work for some time.
Start from the bottom of the stone and work your way up. That minimizes streaking. I generally work in eight- to ten-inch bands (depends on the size of the stone).
While moss is beautiful, it will harm stones in time as it grows rootlets into the stone and weakens it and also retains water which, if it freezes in winter, will damage the stone in weak areas such as carvings and the edges of lettering. Lichen and moss also damage stone by actually using the minerals that compose the stone. Other biological agents can stain the stone such that it is unreadable. So removing such will help preserve a stone.
I hope that's helpful.
Just make sure you have legal standing and permission and proceed carefully.