I have moved into an apartment after enjoying the abundant space of a house. Winter, and now spring, made me struggle with muddy footwear. I discovered that if I put one quarter inch of water in the sink and let the shoes rest there for a short while, the mud softens and I can brush it off straight into the sink then wash down the drain.
Then I put the shoes on my balcony or under the radiator, soles up, to dry. A tiny bit of water and a tiny amount of brushing and clean boots again - no mess on floors of the apartment.
1) That mud you are putting into the drain system will collect and harden like a rock... place shoes in a pan of water, when mud softens, remove shoes and then throw out the water [onto the grass] and hose off the remainder.
2] Heating your shoes will deteriorate the shoes [especially leather], stuff papers in the shoes and let dry naturally, they do make Boot Dryers that just produce slightly warm air, check out the Sporting Goods store [ie, REI].
I have a plastic pan that is a perfect fit for a pair of my husband's tennis shoes. Like Cajun, I put a little water in the pan and soak the bottom of our shoes while the mud is still soft and then pitch the water out in the yard.