There was an attempt at a system in early 2015; it was deemed by State Dept any firearm taken out of the country needed an export permit. Including temporary personal use.
I know because I was headed to New Zealand that year in May; so I jumped through the hoops and had to even obtain an EIN from the government just like if I was exporting 155mm howtizers.(Export Indentification Number). No rental was available, either bring a gun or cancel, so I did what I could to make it happen.
Jumped through some more hoops at a US Customs office at a regional airport where it took most of a full day to obtain a designated transaction number that was filed and all the corresponding forms filled out. Customs agent was great, it was just really tedious to do everything and he did his best, basically I spent all day there. I even went and got us lunch so he could wait for a return phone call and not miss it.
About 2 weeks before I left they dropped the whole thing and you didnt need any of that anymore and just a 4457 for return. So all the time and phone calls were all for naught.
Thats why I kinda chuckle when folks say how hard it is to bring your own rifle. Hard? a few airlines you notify before hand and some not at all, you just show up; sometimes you pay a few dollars.