Based on the coherence theory for non-stationary optical fields, we introduce a new class of partially coherent pulse sources with multi-cosine-Gaussian correlated Schell-model (MCGCSM) and derive the analytic expression for the temporally mutual coherence function (TMCF) of an MCGCSM pulse beam when it propagates through dispersive media. The temporally average intensity (TAI) and the temporal degree of coherence (TDOC) of the MCGCSM pulse beams spreading in dispersive media are investigated numerically, respectively. Our results show that over propagation distance, the evolution of pulse beams is from the primary single beam into multiple subpulses or form flat-topped TAI distributions by controlling source parameters. Moreover, when the chirp coefficient s < 0, the MCGCSM pulse beams through dispersive media will show the characteristics of two self-focusing processes. The reason why there are two self-focusing processes is explained from the perspective of physical meaning. The results in this paper can open the applications of pulse beams in multiple pulse shaping and laser micromachining and material processing.