If the acceleration is not constant, the curvature of the time space is not quadratic but of a higher order, but the same principles apply. Any curvature of timespace can be observed as gravity or acceleration, depending of the frame of reference. It is just that we don't have a supply of massive bodies that loose or gain mass in an observable fashion, hence the textbook comparison is done to a constant acceleration, rather than varying acceleration.
The general theory of relativity is called general because it allows for any frame of reference.
I am glad to learn that massive objects don't disappear into empty space. But I think we should probably prepare ourselves. Art Bell may have a guest soon that tells us that is exactly what will happen when that asteroid hiding behind the moon crashes into earth next month.
It isn't necessary to postulate disappearing mass to see that a reference frame traveling in the space-time continuum of general relativity, represented by smooth curved space, will undergo varying rates of acceleration. But every object in that reference frame undergoes the same varying rate. If the car is in outer space, the cup of coffee on the dash board doesn't slide.
Not all accelerations are due to gravity modeled as smooth curved space. For the car traveling down a road on earth, the acceleration is due to the friction force between the tires and the road. The frame of the car undergoes acceleration, but not the coffee. So the coffee slides. The same would be true of a car in outer space being tossed around by thrusters firing erratically on the outside of the car. The frame of the care would accelerate, but not free objects in the car until they bump into something.
We are talking about two different scenarios. There is no need to invent mysteriously created fictional forces for the car on earth.