Spray the slate with a coat of water. If the sun is out, you may have to do this a couple of times as you move along the path.
Squirt a few drops of dishwashing detergent into a bucket and fill with water. Scrub the slate with a green pad dipped in the solution. Green pads works well on dirt and fungus growth. Only clean a few tiles at a time before rinsing, so no tiles dry with detergent on them.
Rinse all of the tiles when finished. If you are doing a garden path, or there is grass close to the path, rinse for about 15 minutes to dilute detergent so it will not harm plant life.