Hmm, I've got a '78 Ferrari 308 GTB, which is a MR. Never seen any weight shifting in the car! The weight, aside from sloshing of fluids (e.g., gas in the tank), isn't moving. What it is is a change in dynamic forces. Consider going in a straight line at constant speed (acceleration is zero). Contact force at the wheels is constant, and the balance is based on the weight distribution of the car. Now consider the same car suddenly lifting off the throttle. The car, due to the location of the CG above the contact plane with the road, wants to pitch forward. This increases the contact force on the front wheels and reduces it on the rears.
Now resistance to side slip/yaw is from friction at the wheel/road interface. Simple model for friction is that the friction force is linearly proportional to contact force, or F_fric=mu*F_contact, mu=constant. I'm sure most games use such a model, though in reality it's much more complex (i.e., mu is not constant but a function of contact force itself). Under zero acceleration yaw resistance is provided at all wheels. With that change in acceleration and reduction in contact force, friction force at the front increases, but at the rear it decreases. So it takes less side force to induce a yaw motion.
It's as simple as that. Simple dynamics.