My answer may be a bit controversial but when one of the dogs enters the house under those circumstances I yell, “Honey the dog just came in and it’s muddy!”.
On one sit they had a paw cleaner device they bought on Amazon. After the walk you take each paw and individually put it into what looks like a round jar. You move the paw up and down and it cleans the paw. Then you wipe the paw dry. The dog has to be good with this. Usually after paw three the dog had enough so the last one I cleaned the old fashion way. The pet industry is a billion dollar industry for a reason, ha.
At home, we have washable rugs and towels specifically for our dog. He’s learned to let us wipe him, including rolling over so we can dry his underside and paws. We have tile floors at our entry, so that’s easy to clean. We keep similar rugs and towels in our vehicles.
When I sit, if it’s rainy or likely to rain or snow, I ask about towels to dry the dogs. Those hosts have had dog towels as well. One of them had a Swiffer-type mop with a spray feature, which made their tile floors easy to clean.
If a dog needed it, I’d rinse them off and dry them.
We don’t have related photo to share but we’ve come across plenty muddy (and snowy) dogs. We looked after two working cocker spaniels (well trained but unquestionably active). After each of two walks per day - plentiful mud, brooks and mess - then they were thoroughly hosed down using an external tap. Twice per day! Dogs did not mind water hose at all as they were clearly happy
We live in a mild climate in Southern California, so we usually just take her straight to the backyard (we have pavers) and give her a hose down. We towel dry and put on her doggie robe and then she relaxes inside
Mardy loves getting rubbed dry with a towel (he has his own).
He loves a bath as well. When he was a puppy, he had terrible FOMO so we trained him to get in the bath by sitting in an empty bath fully clothed. When he used to jump in himself, we turned the tap on gently and he would get used to running water and now he sits for a bath. But he loves a good towel rub. And lying in front of an open fire! Really is a dogs life!!
I had a sit where we entered through the basement, a beautiful renovated basement. And the laundry room had a big sink for dog washing. Even after every walk, she asked me to put the dog in and clean her feet. Small dog, so easily picked up. But it was very handy. For big dogs, I always ask where the dog towels are. And also, where the tomato juice is. In case of skunks. Some people have a skunk kit. But I like using tomato juice outside, then rinsing off outside. Since you cannot let them in the house, if sprayed. Then I wash them.
Blankets on the back seat of the car and a hose down in the garden when we get back. Fox poo remover shampoo optional depending on determination of dog and reaction speed of sitter.
We are currently sitting a black lab and she loves the beach and the sand, but washing her off is almost a sport on its own. The moment the hose comes out, she’s bouncing around like it’s a game of tag, turning cleanup into one big playful chase.