I brought my trophy back on the plane with me.
I killed a tahr, and two young red deer stags. I brought the tahr's skull and horns and cape back. I turned and salted the cape and hung it in the wind a day. It was not dry by any means and I rolled it and kept salt on it for the journey. I had boiled the skull and removed horns.
This was in Nov 2016.
I went to a NZ F&G type place before I hunted and got some type of permit to help with export. I listed basically any animal I might kill (2 tahr, 2 chamois, red deer, etc) because it would cover whatever. It didn't matter if an animal was listed and I didn't export....it mattered if I tried to export something I had not listed.
I remember filling something out for the US end, but it wasn't a big deal. I was pleasantly surprised how easy it was to get it through customs. I was pulled aside and it was looked at, but no biggie.
Shipping stuff is ridiculously expensive. Sometimes as much or more than the hunt. To date, I've left trophies in Zambia, Australia and Argentina....all great hunts, but decided I liked hunting more than bringing stuff back and am content with the photos.