I am not aware of such a cutting fluid - may be the producer has to recommend if there is. But please bear in mind the following:
1. You will have to circulate the fluid in order to save it (save money as well).
2. After the stone being machined, there is a lot of waste from the stone getting into the fluid
3. The waste has different grain - it depends on the stone - limestone waste is very fine(like mess), white marble waste has a little bit bigger grain and granite grain has biggest grain. Then you will need a separation installation with deposit concrete tanks to filter and collect the waste - otherwise you could have a problems with the sewerage and local authorities.
4. At my work place I use water for coolant and we have a separation system with wastage collectors. Even after being filtered it the water contents some waste of previous machining. That is why you can not use all the same water or other coolant and you will have to renew from time to time. If you find a special liquid you will have to think about the expenses.
5. You can ask for coolant liquid in factories where they have lathes for mashining metals - they usually use a white liquid which do not rust.
6. I don`t think that liquid will colour the stone as it contents mainly grains , not paints. Of course you will have to be carefull when you machine painted granites or marbles such as Nero Zimbabwe, China Black, Verde Gwatemala. Rainforest Brown and Rainforest Green have natural paint - you have to be careful.
Best regards and luck. If you need any help, just write to me - firstname.lastname@example.org