The theory of stationary spatially localized states in dissipative systems driven by time-independent forcing is well developed. With time-periodic forcing related but time-dependent structures may result. These may consist of breathing localized states, or states that grow for a time via nucleation of new cells followed by cell annihilation for a different length of time, all depending on the nature of the forcing oscillation. A phase diagram is computed locating different types of dynamical behavior in the parameter space of the periodically forced quadratic-cubic Swift--Hohenberg equation. An explanation of the phenomena observed is offered, based on asymptotic arguments.