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 also be found if you search here. 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