Typically, a rifle is stood muzzle-up, and hand carried muzzle-up at ranges. But the muzzle is high and not pointed at your head. Some ranges will have rules about that.
When cased, the same rules don't usually apply. Not sure about open carts. I get a queasy feeling when I see someone wheeling a cart with an open rifle, mounted low, with the muzzle pointed at their head.
Try to design it so that the cart never directs an open muzzle at you or anyone else.
Whatever direction the gun is, I prefer a case that blocks the muzzle, especially when moving about other people.
I don't even like to see the common FT carry caddys with a muzzle pointed in my direction.
Or like this?:
It might depend on the circumstances.