Throughout the country, there are canine underground railroads at work transporting dogs (and cats) from one side of the country to another, often from shelters/rescue homes to forever homes. The groups come in as many flavors as dogs come in. Some specialize in specific breeds and others move whatever dog or cat needs a ride. Some are funded by the volunteers that transport the animals and others are more organized, providing more financial assistance for the volunteers.
The basic plan is a daisy-chain of volunteers who drive a “leg” of the dog or cat’s journey. Another volunteer picks the animal up from a pre-determined location — a parking lot, a gas station, a park, etc. — and takes over for their “leg.” This continues until they reach it’s new home, or maybe it’s new shelter.
Below is a partial list of transporters. More can be found if you search Yahoo groups for pet transport. There are many groups you can post your transport needs to or request to be a volunteer.
See also this post for more pet transport options