I think Chillypepper is on to it. It's about an abusive relationship. "She's" running from "Happiness" because whoever it is that she's loving (ie happiness) is hurting her ("happiness hit her like a bullet in the back, struck from a great height, by someone who should know better than that").
The Dog Days were the good times, the horses coming is the impending doom of this abusive relationship. Because the drums, tone, and rhythm of the song seems to invoke that Native American theme I'd say the sounds of horses representing the whites coming to take everything from the Natives, thus ending the Dog Days is an apt observation.
The chorus reveals the message of the song "Run for your mother, etc" is telling her that you can't hold on to an abusive love or it will kill you, go back to those who support and love you in a positive way (mother, father, etc).
Not really a Christian song but as an outsider the basic message of Christianity has always seemed to me as being "Provide and seek positive and healing love to all your fellow man so that we may all overcome the pains of life and in that find and provide salvation to the world". I believe that message can be found in this song.
Maybe I've got it, maybe not but I think we can all agree that this is one great song.