Depending on how soft the muddy ground is, I might consider putting down some cheap plywood so there's at least somewhat of a solid surface to travel over.

As for the hill, how steep? If it's significant, maybe consider a make-shift sled of some sort and pull the game up the hill (assuming the ground is slick with moisture/snow)

Two things I would do with the hand cart--make sure to wrap up your games in stretch wrap to water proof them in case they tip over. Second, use a couple of straps and strap the game to the cart. It's easier to tip a toppled game back into the correct position when it's still attached to a cart, rather than to try to place the game back on the cart after it's fallen off. Also, don't forget some padding on the corners of your game where the straps make contact to prevent damage from rubbing or compression.