I might as well throw my 2 cents in.
The only time CG shifts in a car is when the fuel load is burned off and, by very small amounts, when the suspension deflects and parts are displaced from their static location. It can also move when the driver has an "oh shit!" moment and loses several onces of waste matter.
What does change in every vehicle is what is called the lateral force coefficient center. This is the point on the side of the car you could push against to slide the front and rear axles sideways by equal amounts. What our goal should be is to keep that point from moving ahead of the center of gravity. If the lateral force coefficient center moves ahead of the center of gravity, then the vehicle will become unstable. Any divergence from a straight path will create a moment about the center of gravity which will, in turn, add to the divergence.
There are a HUGE number of factors at play that contribute to the location of the lateral force coefficient center at any given instant. For a good visual of how important that point is just watch a video of a winged dirt track sprint car.