We study a time-varying well-posed system resulting from the additive perturbation of the generator of a time-invariant well-posed system. The associated generator family has the form A+G(t), where G(t) is a bounded operator on the state space and G(·) is strongly continuous. We show that the resulting time-varying system (the perturbed system) is well-posed and we investigate its properties. In the particular case when the unperturbed system is scattering passive, we derive an energy balance inequality for the perturbed system. If the operators G(t) are dissipative, then the perturbed system is again scattering passive. We illustrate this theory by using it to formulate the system corresponding to a conductor moving in an electromagnetic field described by Maxwell's equations.