The Root of the Behavior
This question can really be taken two ways. If you're wondering about new indoor environments, dogs tend to gain a lot of comfort from the ability to mark the new 'territory' with their scent. Depending on the size of the home, this can take anywhere from one week to sometimes over a month. This process can't be rushed, it requires patience and time. As for wild dogs, their adaptations are incredibly varied. The most obvious adaptation tends to be canine fur. You have certainly seen dogs with shaggy fur like a Siberian Husky or a Golden Retriever. But the same species can also be close to hairless, just like a Chihuahua. Incredibly, these dogs originate from the same basic wolf species that we started domesticating tens of thousands of years ago! This is a great example of a canine adaptation based on severe climate variations and seasonal drift.
Another great example of an adaptation is the ears of the African wild dog. They differ from most other breeds in how rounded they are. This shape serves a very distinct purpose, as their howls contain an underlying low-frequency tone that is much better caught by this conical shape. Incredibly, scientists seem to agree that these dual features happened independently of each other, and were originally due to different and distinct environmental pressures.
Adaptations do not have to be obvious either. These same African wild dogs have changed their behavior based on adaptations to nature. In the sparse areas they inhabit, pack unity is incredibly important to survival. Because of this, these dogs have evolved a distinct lack of violence towards each other. There does not seem to be very much seperation of gender, and they have been seen helping their old and sick by bring them food and keeping members of the pack within earshot.
Encouraging the Behavior
Habitat tends to be the strongest motivator for evolutionary adaptations in canines, but the second strongest is almost always diet. Sudden changes in nutrition can affect dogs on a generational basis, instilling new and minute differences every time. After 20 or 30 generations it then becomes possible to see adaptation at work. If your pack is constantly short on food, perhaps 20 years down the line the average size of these dogs will be smaller, and more adapted to food shortages.
Another type of adaptation can happen in behavior, more specifically in hunting behaviors. Say for instance a pack of dogs ends up occupying ground next to another pack of wild animals. Normal hunting behaviors wouldn't work, as dogs tend to pack hunt purposefully to single out individuals prey, and an opposing pack wouldn't leave members alone. This means that a new strategy would have to be developed in order to survive, thus forcing their 'biological hand' as it were. Yet another type of adaptation can be specific to individual domestic dogs. Their brains are uniquely adapted to take social cues from humans. This means that they can learn to change their instinctual behaviors, or even learn completely new ones based on what you teach them.
Other Solutions and Considerations
Dogs that have moved once or twice tend to handle a change in environment a little better than freshly adopted strays or wild dogs. They've been raised to be confident in shifting situations, and can handle being put somewhere new. One thing that most animal behaviorists warn against is medicating your dog in these situations. Anxiety is a normal response in a lot of animals dropped into new environments, and needs to take its course before the average dog can settle into a new space. Training can always aid in this process. Professionals in the field will oftentimes have a larger array of tactics at their disposal.
It's pretty clear that dogs possess a crazy amount of adaptability. They can change their bodies to suit a variety of climates, adapt the tonality of their howl, and even modify their social behavior to better survive in the wild! Thanks to biology and human intervention, we have seen all sorts of neat changes. Best to just enjoy the colorful array of breeds we've all come to know and love!