Look at HG specialist products to remove dirt and old polish then start again with one of their protective coatings. See
this and
this. We used their stuff for years on our slate floor in a previous house and it really is worth the time and money. I never found it a problem with the dogs as the more aggressive cleaning products are all finished with fresh clean water.
Once you've got a good finish on the floor you can simply touch up worn areas as needed to prevent staining and just use their normal
shine restoring cleaner* I promise I don't work for HG but we've had a lot of natural stone floors over the years and every installer has raved about, and recommended the HG range of products...
I'd also suggest using a Turtle Mat near your back door to absorb water and muddy paw prints - I've found them worth their weight in gold - a quick check should bring up a lot of old threads about these together with some alternative suppliers x