What is the best way to mop a slate floor? What cleaning solution should we use? We have slate tiles in the kitchen with concrete mortar. You can see from the pictures that the surfaces of the slate tiles are uneven. They are also porous (water dripped on them is absorbed). I sweep regularly, which isn't easy because of the uneven surfaces. I have mopped once with a Swiffer Wet Jet but I'm not sure whether that's OK to use repeatedly.
Our kitchen slate floor is similar to yours. I use a damp mop -not soaking wet- with a bucket of Pinsol. In between deep cleaning I use the Swiffer Wet Jet. Our problem is the grout between crumbles and has been replaced 3 times. Every time new grout is put in - I seal it with a sealer from Home Depot.