In this paper, the propagation of Temporal Coherence Grating (TCG) pulse trains in a dispersive medium with a chirp is investigated for the first time. The two-time mutual coherence function of the TCG pulse trains propagating through extended dispersive medium specified by temporal ABCD matrix is derived and the evolution of their mean intensity and temporal degree of coherence (DOC) is explored. It is shown that the distribution of the mean intensity can be modulated freely by the number of grating lobes N, grating constant a, pulse duration T0, power distributions vn, group-velocity dispersion coefficient β2 and the medium chirper s. Upon dispersive-medium propagation, the single pulse splits into N+1 subpulses with the same or different peak intensities which depend on power distributions vn. What's more, during the propagation the pulse self-focusing occurs being the chirp-induced non-linear phenomenon. And the distribution of temporal DOC will degenerate into Gaussian form from initial periodic coherence distribution with increasing propagation distance z or adjusting incident pulse parameters and medium dispersion. The physical explanation and numerical illustrations relating to the pulse behavior are included.