I know several people who have adopted dogs from southern Europe, like Spain. Those have been adult dogs and puppies, and all of them have adapted just fine. Especially easier if bring them in during spring or summer, of course.
I also took my furry friend to the humid temperatures of East Coast in the States, and she adapted just fine, even though was used to much less heat here in Finland. Just had to make sure not to have walks around midday (lots of fur..), which was the reasonable thing to do anyway - and that's how the locals with dogs did it, too.
A funny example about puppies & doing their business in the snow:
I had a puppy, who was born late December, and it was one of those really really cold winters (-28 C day temperatures in February) in central Finland.. So the lil guy was not used to the outdoors until he moved to his new home, 8 weeks old. At first he didn't want to pee outside, as it was so friggin' cold, but very soon got used to it. The thing was, though, that when the snow finally started to melt, the poor guy who was used to "the snow always being there as long as he knew", fought so hard to find the last small unmelted bits of snow in the parks or roadsides, which were the places where he used to go poop.. not on the already melted grassy areas.

Yeah, he got over that in a few days, but before that I did see him struggle to find the last snowy areas, as that is what his mind was used to. I guess this is sort of the opposite of some people never seeing snow before - the dog had never seen the world without snow.
That said, I have never head of any dogs not getting over their unusual circumstances, to the point that the OP wrote about, that a dog's health would deteriorate to that degree. Usually nature wins, and they continue life as normal when they adapt to the new surroundings - that's the good side of living in the moment. So I wouldn't worry that much about it. Dogs adapt easier than people.

Joha mie sanoi, vaikken mittää virkkant.