Dun not only gives a horse a dorsal stripe, but it commonly lightens the base color and may also give the horse zebra striping on it's legs, shoulder barring, and sometimes lacing on the forehead.
I guess the real reason I got to really thinking about this is the herd of feral horses we just rounded up in the last couple of weeks. There were many of them that had dorsal stripes, but I know for certain that there isn't, nor has there ever been, any dun genes in their breeding pool.
Unfortunately, I only have decent pictures of 2 of them with prominent dorsal stripes, but I think it will be enough.
Oh, and please, no comment on their condition, they aren't my horses.
Here is an example of a chestnut horse with a dorsal stripe due to sooty/countershading.
Here is a random internet picture of a couple of red dun horses
Here is the second of the feral horses. She has a bay base and the sooty is clearly evident on her neck and shoulder area.
And here are a couple pictures of her dorsal stripe
As opposed to these bay dun horses
Anyway, just showing that a dorsal stripe doesn't automatically mean that the horse is carrying the dun gene.