There are special swede brushes and swede stain erasers / blocks on the market. Any shoe specialist or even the shoe repair man should have these products.

I don’t have any similar shoes, but I would suggest wiping with a damp cloth. Should remove most of the mess

Id say with just a damp cloth

Yes I agree with contacting a shoe store for sure!

Could it be suede? If it is, you can get suede cleaners and also a protectant spray to keep them clean.

I would suggest contacting a shoe store for advice or even take the shoes in and ask them face to face – worth a try.