Firearms - Canada to USA to Mexico and return ?

I'm considering a big game hunt in Mexico next year. I would drive from BC to Arizona. Importation of my rifle into USA would be covered by my BATF permit. Taking it into Mexico would be covered by a Mexican import license provided by my guide. However, I'm not sure about exporting from the USA and re-importing from a country other than Canada.
Can anyone shed some light on this ?

Thanks,

Would not your ATF permit cover you both ways?

I would just call the ATF. They have the answers that you are tying to find, and it would be up to date.

Shoot Joel a email at www.borderview.com I’m sure he can set you in the right directions with some answers ... he’s a good guy

Your BATF permit is expected and honoured when you return to the US from Mexico although after at least 6 such trips I have never had my firearms inspected nor documentation requested when landing in Phoenix, enroute to Canada.

The atf application is a bit confusing. Can anyone shed any light on this? It asks if you have an immigrant visa then if you say yes you supply the hunting license.

Im going to New Mexico this fall for elk and wondering if I need to submit my hunting license with my app.
If someone who’s has gone through these motions could respond that would be great.

thanks
chuck