Check your airlines rules on extra baggage and antlers, this is critical. Know that elk is way different than mule deer or antelope when flying meat/antlers.
Here is what I do:
1. Buy a small blue/red igloo cooler from Wal-Mart for $20 when I fly in as it holds 43 lbs of meat and 5-10 lbs of dry ice perfectly. Buy more as needed if you are succesful and can can ship meat back. Other options are buying or making an insulated box, using a duffel with insulation and garbage bags that weigh a bit less....but I like the cooler myself. The key is to find a light cheap cooler and Wal-Marts are the ticket as they only weigh 6-7 lbs so you can pack 43 lbs of meat. I load it up and weigh it and get it to 49 lbs and add dry ice and throw out the dry ice when I check it in. You need some sort of strap or two like a canvas belt strap to go around the cooler. If you have less meat per cooler, you can keep up to 5 lbs of dry ice in the cooler, just have to declare it when you check in. Typically a block of dry ice is 8ish lbs and by the time I get to the airport it is less than 5.
Antelope buck = 40 lbs of meat (one cooler)
Mule deer buck = 50+ lbs of meat (one cooler plus)
Bull elk = 150+ lbs of meat (3 coolers)
The best airline for flying and shipping meat is typically Southwest as they are $50 per extra bag.
Elk horns are a pain to deal with on some airlines so you might want to ship them somehow. If you are going to mount a mule deer and you are shipping back the cape, you could cut your skull plate at an angle or some sort of Z to cross the skull with a hack saw and mesh the antlers in a duffel bag easily, as long as you are not measuring it for B&C. Remember to cover the horn tips with some sort of garden hose and tape so they don't puncture your bag.