TEX: First, check airline policies, if you're flying. It's a hassle, and you better have an aluminum hard-case.
If driving south of the Rio Bravo, first get all paperwork from a Mexican Consulate. You'll go to jail for even a cartridge, much less a gun, if you don't have all the "permisos". Also, Mexico does not, repeat, not, allow any military cartridges. No .45ACP nor 9mm. No .308 nor '06 nor .223. At all.
When thinking Mexico and guns, think "JAIL" unless you have the paperwork.
Costa Rica is cool, unless laws have changed. I saw an ad in the "Tico Times" for a Ruger Blackhawk, .357...And the rice farmers love for hunters to shoot those nasty white-winged doves which eat the crops.
And, yeah, check the NRA site...
FWIW, Art