Is it safe for the Kentucky Derby horses to run on that muddy track?

A well cared for muddy racetrack is much safer than a poorly manicured dry racetrack. The mud makes the horses run slower... it's like when you're running and your feet keep getting stuck in the mud... it slows you down. But there's nothing dangerous about it. Some horses like running in the mud, others don't... that doesn't mean that any horse is going to be injured because of it... it's just that some horses aren't going to run as well as the others because of it. The racetrack knows when conditions are unsafe and they'll cancel racing if they're concerned about injuries... that often happens during the winter snow storms or when there's serious flooding that they can't drain off the track. The Churchill Downs crew are caring for the track very well... they squeeze out the water with heavy tractor equipment between races, so the horses are running on the dirt, not a soupy mess.