It depends on what kind of game animal you are transporting and how much meat/horns we are talking about.
If ~40 pound of meat (doe, a few turkeys, etc.), then freeze the meat, wrap your packages in freezer grade aluminum foil, carefully back in a light backpack or duffle and take with you as a carry on. If you have a rack, then remove all hide and as much flesh as possible and wrap with thin cloth and strap to your hunting pack/backpack. It is legal to carry horns/antlers on an airline and I have done it several times. Great conversation starter. Don't know what I'd do about elk...
If you have a long layover or flight, you can add a layer of bubble wrap from Wal-Mart, though this stuff is somewhat expensive. I still use it in situations.
These recommendations assume you have processed and frozen the game somehow. For big game, I'd prefer to get it solidly wrapped in plastic game sleeves you see most game processors use or vacuum packed. For game birds, you can get away with a few gallon ziploc bags. In both cases, if you freeze the meat it will still be frozen (mostly) when you get home.
I don't bother with dry ice, opting instead for extra insulation. I don't care if edges of my meat defrost. It's totally okay to refreeze.
Other suggestions were good too.