The Trail Nerds trail running group in the Kansas City area puts on trail races and welcomes dogs. They sometimes include them in the race results.
There are also a few dog themed road races for animal rescue charities where dogs are specifically invited. It seems strange to me that such races are so frequently scheduled in the hot summer months. There are many breeds that are just not suited for sustained running in hot weather and though I have never been to any of those events, I would have to assume that some people bring dogs who are not conditioned for the event.