A theoretical approach is developed to solve the problem of nonstationary elastic light scattering by a dielectric spherical particle. This approach proposes optical fields of the scattering radiation to be represented as an expansion in terms of eigenfunctions of the stationary problem, in which the expansion coefficients determine the temporal behavior of the field and comply with inhomogeneous oscillation equations. Transient processes at formation of optical fields in a microparticle are studied. It is shown that nonstationary pulse scattering manifests itself, first of all, in the time shift of maximum of the internal field relative to the profile of the initial pulse and in the time delay of its trailing edge. This behavior of nonstationary fields is connected with the resonance character of the process of elastic light scattering by a particle, in which vibrational eigenmodes of the internal optical field with lifetimes comparable with or much longer than the laser pulse duration are excited.