The cars are moving even though they are not creating the energy - it is transferred to them. Therefore, they both have kinetic energy.
The short answer to your question is yes, they do. The explanation of that answer involves relative frames of reference. Kinetic energy is the energy of a moving body, and is usually calculated as KE = 1/2 mass x velocity^2. In the case of the cars, each would have kinetic energy equal to 1/2 of the car's mass times the car's velocity squared, regardless of how the velocity is being reached.
Whether something is moving and how fast it is going is actually a complex question, since the entire universe is expanding and therefore everything in it is moving somehow. In order to simplify physical mechanics, the surface of the Earth is generally regarded as a fixed plane for the purpose of calculation. The common physics convention is that you should use the Earth as a fixed frame of reference unless there is a stated reason to do otherwise.
I think your examples have to do with relativity. Kinetic energy is defined as the energy of motion. Relative to the transport, the car has no motion, however it is moving if the ship is moving. The car could also be viewed as having potential energy relative to the transport because if the ship stopped suddenly, the car would move.
Relativity is needed to decide if anything is moving or not. The Earth is spinning and orbiting the Sun, the Sun is spinning and orbiting the central black hole of the Milky Way Galaxy and so on. One could go in a micro direction as well.