It's easier to remove mud from running shoes when you let the mud dry first. Then you simply put on your running shoes again, stomp your feet, and the dried mud should fall off. Or hold a shoe in each hand and knock them together with a clapping motion so the impact shakes the mud off.
Do your running shoes look nice again? If they’re still muddy, you can spray them with a garden hose, or hand-wash them by unlacing them completely and using a soft brush and a mild detergent. If the stains simply won't come off, place the shoes (and nothing else) in your washing machine. If possible, remove the sock-liner or insole and wash it separately. Laces can be washed or replaced.
Let your shoes air dry. The heat from a dryer will break down the adhesives that hold your shoes together. Don't place them near a heating device such as a radiator since direct heat will alter the shape of the shoe. Open up the shoe and stuff the inside with paper towels or newspaper to absorb the dampness inside the shoe.
Running shoes take about twelve hours to dry out. If you're a frequent runner, you may want to invest in two pairs of shoes for running so you can alternate between them as each pair dries out.