I've dogs from Cyprus in the past. They had full health clearance, full passports, jabs, all required treatment, etc. They had been in foster homes, who had made good starts on training & teaching them to live in a house.
You can't get them here before 16 weeks, I think it is? As they can't have the rabies jab before 12 weeks, and then have to wait a number of weeks after before being cleared to travel.
The rescue we used (and it is a rescue, not a business, based on a pound) does send a questionnaire, talk to you thoroughly, and also do a home check, and they do have some rescue backup in this country.
Our dogs have been wonderful, but very different from each other. And a fairly big learning curve. I have seen others struggle more than we did, but that seems to be mostly with those who have taken on older street dogs, rather than pups. I think being in a foster home in their native country before flying helps too.
Would I do it again? I'd certainly consider it. Our reasons for doing it were sound, and if the circumstances lined up that way, I'd go that route again.
I'd also say it's not a cheap route to get a rescue dog - we ended up paying several hundred pounds per dog, which covers the flight.