I'd need to know a bit more about your winter hiking conditions and duration.
If you're erring on the side of active, I'd suggest a Labrador.
If you're doing colder and shorter, something like a Bernesse Mountain Dog would do amazing.
I love running with my Labrador since she can handle heat decently
(I don't run with her when it's very hot), she can handle cold better than I can (has had her whiskers and chest get ice covered), and can handle decent distances in comfort (6-10 miles consistently to age 9).
My only real tips for running with any dogs - be considerate of the heat and make sure you have some good water sources for them.
Update (after 2 years):
Our Labrador, now 12, has some hip issues
so she's limited to running an easy 2 miles.
I never considered Western medicines in the past but acupuncture seems to work wonders for her.
We also adopted another dog which we believe is a Lab / Catahoula mix and, holy crap, can that dog run.
I have never heard of Catahoula's before
but she has no problems keeping up on 15-mile mountain bike rides.
My trail runs don't tire her out unless I'm doing 10+ miles, which means she's probably doing 2x that.
I'm glad to see them on the list but wouldn't have put them under long slow runs!
As far as running, this article on Runners World should answer; Ref: The Best Types of Dogs for Runners
outdoors.stackexchange.com/users/4135/mattsolar
Australian cattle dog or other types of working dogs.
Mine loves backpacking.
Cattle dogs have very high energy levels and are bred to travel long distances.
Excellent for trail running and hiking long distances.
They can carry their own food/water and will do so without whining or stopping (working mentality).
They have a very active mind and are very alert, good as a watchdog but have very sweet temperaments.
They require very little in terms of grooming.
They have a short to medium-length double coat which is naturally rain-resistant but also can be dried quickly.
They are medium-sized, in case you need to carry it if injured.
Very trainable if you're looking on taking it out off-leash.
Also to note, if you're going to get a long-haired dog for backpacking in the summer, be sure to have its hair cut very short.
It's a myth that hair keeps dogs cool--
they get really hot really fast.
I also have a long-haired Australian Shepard that goes summer camping in Texas weather with its summer haircut.
outdoors.stackexchange.com/users/3066/Chris Mendez
Somehow, I can't see a Corgi keeping up.
I have a 20-pound terrier mutt.
She does great with me trail running at distances of 6-7 miles.
After we get home, she runs around the back yard in circles like a rocket.
Dogs are just much more efficient runners than humans, especially in cool weather.
As far as I can tell, humans only seem to be at all competitive with dogs at distances more like 12 miles.
(I haven't taken my dog that far, but have seen my running buddy's larger dogs act tired at that distance.)
There is a theory that humans evolved for a hunting niche on the plains of Africa, in which we were specially adapted to pursuing prey for long distances in hot weather, but we're talking about really long distances.
For hiking, the problem I've found isn't really my dog's ability to do it physically, it's that she tends to injure her feet.
You can get various kinds of booties.
It's hard to find ones that fit correctly and that stay on.
Keep in mind that if your big dog hurts a foot in the backcountry, you're going to have big problems getting him out.
Mine I could carry out if I had to.
outdoors.stackexchange.com/users/2169/Ben Crowell
The wolf, the wild ancestor of the dog, has extreme running endurance.
According to this article , many wolfs travel more than 50 miles daily searching for food.
I've read somewhere, that wolfs can chase moose for a few days, but I can't find that now.
Wolf is very similar to us in that domain, human hunters can also run for days.
This can be another reason that two species have formed so special relationship so early.
Looking for great dog runners, look on dogs that are the most similar to their ancestors, such as the German Shepherd (in my country, this race is called Wolfie).
If you choose Husky, you can even count for sleight transport if you are tired.
Probably it would be you who would have problems to cope with the distance.
outdoors.stackexchange.com/users/557/Danubian Sailor
A 3 to 7-mile trail run every day is a bit for most dogs.
It wears down their cartilage over time.
I wore out a couple of labs doing fewer miles than that.
At age 8 they had joint problems.
I just stay with 1 mile walk a day now.
I would consider a dalmatian coach dog.
It is bred for running with the coach.
They are also bred to accompany, protect, and warn.
outdoors.stackexchange.com/users/6728/paparazzo
Which dog breeds are best for backpacking?  I know malamutes do well in the cold and on mountains, but what if I'm backpacking in the summer?   Which dog breeds are good for daily trail runs?   I would love a dog that would wake up with me every day and go on a 3 to 7-mile trail run. Somehow, I can't see a Corgi keeping up.
Dog Breeds for Backpacking and Trail Running
outdoors.stackexchange.com
<backpacking><running><dogs>
outdoors.stackexchange.com/questions/4548
outdoors.stackexchange.com/users/767/theJollySin
Wolf-like dogs, such as German shepherds or huskies. Australian cattle dogs, Bernese Mountain dogs, Dalmatians and Labradors are also suggested. Wolves run well, working dogs are a good option and labradors handle heat well.
Wolves run well, huskies can carry sleighs. Working dogs are a good option. Labradors handle heat well.