Transient electromagnetic interactions on conductive dielectric scatterers are analyzed by solving the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) surface integral equation with a marching on-in-time (MOT) scheme. The proposed scheme, unlike the previously developed ones, permits the analysis on scatterers with multiple volumes of different conductivity. This is achieved by maintaining an extra temporal convolution that only depends on permittivity and conductivity of these volumes. Its discretization and computation come at almost no additional cost and do not change the computational complexity of the resulting MOT solver. Accuracy and applicability of the MOT-PMCHWT solver are demonstrated by numerical examples.