We investigate radiation-imprisonment (RI) phenomena in a system of cold atoms trapped in magneto-optical traps (MOTs) when cooling lasers are switched off for a short period. The problem is formulated within the framework of the Streater-Cooper-Sandle equation that belongs to the Markov class and accounts for photon frequency redistribution in multiple scattering processes. On the basis of a semiclassical approach, we derive explicit analytical solutions for the spectral problem of the RI equation neglecting Doppler effects. The analysis reveals peculiarities in emission profiles under initial partial saturation of the resonance transition by the probe laser pulse. Comparison between theoretical and experimental data on effective radiative lifetimes obtained in Cs and Rb MOTs confirms the validity of the discussed nonstationary model of RI in cold samples. For small detunings, the Streater-Cooper-Sandle equation agrees with the experimental data, while in the case of large detunings, the theory does not agree, probably due to non-Markovian effects.