We report theoretical studies of the resonant scattering of light by perturbed atoms under conditions where the amplitude of the incident electric field varies with time. Adiabatic perturbations are simulated by a randomly fluctuating term in the energy separation between the resonant levels; lifetime effects are accounted for by adding a small imaginary term to the separation. Assuming linearity, we derive a general expression giving the intensity of the scattered light as a function of time. Explicit calculations are carried out in the motional narrowing limit for a variety of cases. Interference and inhomogeneous broadening are assessed. A brief analysis is made of recent scattering experiments involving Cr3+ in Al2O3. Incoherence in the source and the motion of the target atoms are also considered.