I think most governments would want to keep excavated artifacts of historical significance within their borders, permit or not. No government is going to give y

I think most governments would want to keep excavated artifacts of historical significance within their borders, permit or not. No government is going to give you a permit to take their region's history from them. I'd expect the litmus test to broadly be "if you took something of value, that's bad". Whether it's illegal to own one of these artifacts is going to vary. Some countries have laws that make it illegal, but many don't. It's usually always impossible to legitimately sell these artifacts, since you can't show provenance (and if you could, the provenance would show that they were stolen). Just like stolen art can't be sold—even after many years and owners—trafficked artifacts can't be sold either.