Gentlemen this is one area of cleaning you really have to do your research on,get it wrong and your in a world of hurt.
I'll start 1st with the subject of cleaning marble and granite,one of the best ways is with.........
Baking soda

yes baking soda
Rub baking soda into the marble with a damp cloth. Then, rinse with water; towel dry; and you're done!
Baking soda is a light abrasive and a natural disinfectant, so it's tough on stains
why use it?
its cheap for a start
its a sanitizer
good stain remover
won't etch marble
DO NOT USE vinegar, lemon or other acidic stuff to clean marble – this can lead to etching
The lettering,this can be made up of anything from lead slate or the gold stuff i wont go into this alot but do your homework(google) on it as this is the most important part.
For other headstones you really have to find out what the stone is,some are sandstone,limestone,the list go's on you can get away with pressure washer the harder stone but for god sake not the softer stone.
OK,why do i give this warning,i like some of you thought about doing this a few years back,got asked to do one and without any knowlege slightly buggered up the top edge of the headstone using to much pressure beleive me the grief i got from another family member

It really is one very delicate bit of work to do,think about it,if someone came along and damage your family shrine......................... how would you react?