I don't do a lot of traveling in the rain. If I have to, I tape the muzzle, or cover it with saran wrap and a rubber band, but still carry it UP. I stand in the rain a lot, and that is when the gun is under my poncho, with the muzzle down. That keeps water from running down my stock and getting into the barrel mortise, and lock mortise.
I realize other member's experience is different than my own. I have explained my choice based on safety issues. I found out, the hard way, that carrying a gun muzzle DOWN when you are moving, is a great way to plug that barrel with mud and leaves, if you stumble and fall, or slip. I have accident reports where someone's gun fired into the ground, and a projectile ricocheted off a hidden rock and injured either the shooter, or someone near him.
Carry your gun as you want. I only hope you don't ruin a hunt, or worse, by having an avoidable( ie. Preventable) accident. They don't happen often, but its always a shame when they do occur.