I went armed with a spray bottle filled with soft Sheffield water to dislodge ground-in/weathered dirt and/or bird muck on my relatives' gravestones, which were both the stone type and granite ones. Worked a treat. Some patches needed softening with a first saturation squirt, then a wait, then a second few pointed squirts to loosen. If I could have got a power source for my Karcher I would have taken that too lol. Alas just elbow grease available. Older stone is the worst with moss/green staining under trees etc etc. I used a medium brush on those after softening. Granite was a breeze. Just wipe off after spraying. However I noticed the gold painted indented lettering had fallen out with age and weathering, so there is a lesson there. Don't pay to have gold lined indented lettering as it is a waste of money if it falls out after a year or two.
Somewhere I have a booklet regarding cleaning but can't remember where I purchased it from as it was many moons ago.
One of my bug bears though is that when Sheffield Council did the safety test on them they laid them flat if unstable.....name side down though so even though there are some of my family graves that have stones, unable to read the inscription.
Regarding spraying though would using one of those garden sprayers be ok to use?
Re laying down unsafe gravestones face-down, the cemetery workers responsible are probably DNA matched to the supermarket workers who put massive yellow reduced stickers over important cooking instructions!! The ones you can't peel off easily.