Definitely something rubber to wear out to the run. I have a pair of these clogs:
Ordered them from: http://www.gardeners.com/Classic-Garden-Clogs/ClogsShoesBoots_Cat,11922,default,cp.html
And I LOVE LOVE LOVE them! They have this thick cork base so they are not flimsy and you can walk in them, garden in them and hose them off easily.
Then get a boot brush or some kind of door mat that has thick bristles to get all the muck out. I leave it right outside the gate to the run. And then slip out of my clogs at the back door.

Ordered them from: http://www.gardeners.com/Classic-Garden-Clogs/ClogsShoesBoots_Cat,11922,default,cp.html
And I LOVE LOVE LOVE them! They have this thick cork base so they are not flimsy and you can walk in them, garden in them and hose them off easily.
Then get a boot brush or some kind of door mat that has thick bristles to get all the muck out. I leave it right outside the gate to the run. And then slip out of my clogs at the back door.