In the context of electromagnetism and nonlinear optical interactions, damping is generally introduced as a phenomenological, viscous term that dissipates energy, proportional to the temporal derivative of the polarization. Here, we follow the radiation reaction method presented in [G. W. Ford, Phys. Lett. A 157, 217 (1991)], which applies to non-relativistic electrons of finite size, to introduce an explicit reaction force in the Newtonian equation of motion, and derive a hydrodynamic equation that offers new insight on the influence of damping in generic plasmas, metal-based and/or dielectric structures. In these settings, we find new damping-dependent linear and nonlinear source terms that suggest the damping coefficient is proportional to the local charge density and nonlocal contributions that stem from the spatial derivative of the magnetic field. We discuss the conditions that could modify both linear and nonlinear electromagnetic responses.