rhrxjehxgglw I got my 5kg dog in Vietnam and have flown her to US, Thailand, Switzerland, Mexico, random domestic US flights. We've slowed down traveling since 2020, so this info might have changed, but it might still be helpful.
Afaik there's no single website/service that lets you filter by how easy it is to import an animal. I found I needed to change my mindset from "let's goโฆ. here!" to "alright here's a list of top 5 places, let's check what the import/export process looks like for each"
Yes, export. You'll have (at least) three sets of paperwork to complete export papers for the country you're leaving, import papers for the destination country, animal paperwork for the airline(s), and possibly transit visa paperwork for any connection countries your flight might stop in. There are fees associated with all of these, and the airline fees can be outrageous.
Bear in mind that the airport agents don't always understand the requirements, or might misunderstand what you're trying to do, so you have to be extremely sure you've ticked all the boxes yourself. We were transiting through China and the Chinese agents took us aside and started talking about quarantine. I said no, no quarantine, because we leave in 3 hours. They talked amongst themselves and ultimately figured out "oh transit" and let us go without any issue. Stay calm and friendly, show your paperwork, and show them their country's requirements if you have internet access.
Asia-based airlines tend to be the most strict. It seemed like it was standard to either disallow all in-cabin animals or cap the weight limit to 6kg (including kennel). My 5kg dog + 1kg kennel squeaked by. They DGAF about emotional support animals and there's a good chance they'll laugh at any service animal paperwork for unseen disabilities. When I asked a Thai rep about service animals he was like "what, are you blind? No? What do you mean service animal? That's not a service animal"
Islands suck. Both as final destinations and as transfer locations. Taiwan, for example, required a full vet workup and transit visa even though we weren't leaving the airport. Australia and New Zealand have strict, expensive quarantine requirements to enter the country (min 10 days, ~$120/day, not allowed to visit the animal during that time) that just weren't worth it for me.
Bring Fido (Google it, canโt post the damn link) was a surprisingly good resource for finding airlines. Always verify their info with the specific airline directly, but itโs a good place to start. When booking flights, if your flight has multiple carriers, you need to call each one and ask what their process is for bringing an animal on board. Ask if they honor the originating airline's policies or if they have their own
My best advice is to pick a destination, research all you can about their import requirements, then go to the best-rated vet in a major city and ask if they can help you prepare your pet and paperwork for import/export to X country. If you're not based in a major city, call the place ahead of time and ask if they have experience with preparing pets for import/export
You'll almost definitely end up "over-vaccinating" your pet along the way. Different countries have different standards for vaccines, and you have to play by their rules if you want to enter. Going from US -> Mexico, for example, I had to re-vaccinate my dog for rabies because Mexico only recognizes the vaccine's validity for one year even though the US recognizes it for three years. In the US, she still had 18 months before needing the booster, but according to Mexico, she should've been boosted 6 months prior
This is all I've got off the top of my head, I'll update if I think of anything else. Hope it helps!