It is the electrostatic field of the charged object C which causes the mobile charges to move. As the charges in the metal separate, the resulting regions of induced charge on the surfaces of the metal container create their own electrostatic field, which opposes the field from C. The field of the induced charges exactly cancels the field from C throughout the interior of the metal. The electrostatic field inside a piece of metal is always zero. If it was not, the force of the field would cause more motion of charges and more charge separation, until the electric field became zero. Once C is well inside the container, almost all of the electric field lines from C strike the container surface. The result (proved below) is that the total charge induced on the inside of the container is equal to the charge on C.
