Wear gloves which keep your hands relatively clean. Then take off the gloves after opening your bag, but before handling the camera. Clean your hands on a bar towel. Then you can handle the camera keeping it relatively clean. I put the camera in a clean plastic bag inside whatever protection I used for carrying it, so it is easy to get out without getting mud on it, and you have somewhere clean to put it down.
After the trip I wipe off all the mud and dust using baby wet wipes.