Drive performs this function in English, it can be used both for persons
2 a : to operate a vehicle
// learned to drive at the age of seventeen
b : to have oneself carried in a vehicle
// drove to Canada as a kid
(source: Merriam-Webster)
and for the vehicles themselves:
- To move along or advance quickly: We could hear the trucks driving along the highway.
(source: The Free Dictionary)
For some persons/vehicles, ride works as well:
1 a: to travel on
// ride a bike
// ride the bus
(source: Merriam-Webster)
but, as Tᴚoɯɐuo notes in the comments, we do not ride a car; we ride in a car.
Your exampleexamples would become:
Imagine a car driving down the way, and then it stops gently.
I'm driving/riding on the way now.
Imagine a car driving down the way, and then it stops gently.
Words like "hitchhike", "trek", "swan off" have very specific meanings and are never used in a general sense.