So much depends on what the specific conditions are. Someone else mentioned socks with ruffwear outers, and that also worked well with me ski mountaineering with my dog in the Cascades. In the winter I also had to keep his feet coated in musher’s wax, and vary layers based on conditions. But setting out with a tin of musher’s wax, socks, and outers for the paws should let you handle most scenarios. Musher’s wax is a good fallback as well in the event a booty is lost and you need temporary coverage for a quick run to shelter.