A Gravestone Cleaning Kit
Of course, you could put together your own kit. Some cleaning solution, brushes, a little pick to pluck mould/ fungus out of the small places, and something to keep water in. But, it supports others if you choose to buy the kit. Also, you won't have to look for the best cleaning solution and brushes yourself.
I think you might bring along something to sit on too. Also, I'd consider an old towel or blanket to spread on the grass. Just in case the grass doesn't like something in the soap/ cleaner you use.
If you pick out the fungus and mould consider collecting all of those little living things and moving them to a new location. I've seen people making gardens with mould and fungi collected from the street.
Of course, bring your camera! If nothing else, get some before and after photos.