|Small amplitude oscillations in prominences are known from long time ago, and from a theoretical point of view, these oscillations have been interpreted in terms of standing or propagating linear magnetohydrodynamic (MHD) waves. In general, these oscillations have been studied by producing small perturbations in a background equilibrium with stationary physical properties. Taking into account that prominences are dynamic plasma structures, the assumption of an stationary equilibrium is not realistic, therefore, our main aim is to study the effects produced by a non-stationary background on slow MHD waves which could be responsible for prominence oscillations. Assuming that the radiation term is proportional to temperature and a constant external heating, we have derived an expression for the temporal variation of the background temperature depending on the imbalance between heating and cooling processes. Furthermore, radiative losses together with parallel thermal conduction have also been included as damping mechanisms for the waves. When temperature increases with time, the period of slow waves decreases and the amplitude of the velocity perturbations is damped. The inclusion of radiative losses enhances the damping. When temperature decreases with time, the period of slow waves increases and the amplitude of velocity perturbations grows while, as expected, the inclusion of radiative losses contributes to the damping of oscillations. There is observational evidence that in different locations of the same prominence, oscillations are damped or amplified with time. This temporal damping or amplification can be obtained by a proper combination of a variable background temperature, together with radiative damping. Furthermore, decayless oscillations could also be obtained with an appropriate choice of the characteristic radiation time.