We study the propagation and damping of small-amplitude prominence oscillations invoking steady flow and radiative losses due to Newton's cooling with constant relaxation time. We find that the strength of steady flow has a large influence on the propagation (e.g. period, phase velocity) of wave modes. In the presence of steady flow, the thermal mode is a propagating wave and hence it can be observed in solar prominences. The thermal mode contributes to the non-thermal line broadening in the solar atmosphere. The steady flow does not affect the damping time of the wave modes. The damping of slow and thermal modes is highly dependent on the radiative relaxation time. The thermal perturbation, in the presence of steady flow, is found to be larger in the case of the thermal mode than in the slow and fast modes. The energy flux (~300 W m−2) associated with the thermal mode is sufficient to heat the quiet regions of the Sun. The slow mode contribution to non-thermal broadening has been estimated. The non-thermal broadening is found to be large in the case of the prominence with large characteristic length. The steady flow, in the presence of Newton's cooling, breaks the symmetry between the forward and backward propagating modes. No modes with negative energy have been found. For strong flows (above 10 km s−1), the canonical backward wave propagates in the forward direction, which can play an important role in wave detection and prominence seismology.