The snapshot colored compressive spectral imager (SCCSI) is a recent compressive spectral imaging (CSI) architecture that senses the spatial and spectral information of a scene in a single snapshot by means of a colored mosaic FPA detector and a dispersive element. Commonly, CSI architectures allow multiple snapshot acquisition, yielding improved reconstructions of spatially detailed and spectrally rich scenes. Each snapshot is captured employing a different coding pattern. In principle, SCCSI does not admit multiple snapshots since the pixelated tiling of optical filters is directly attached to the detector. This paper extends the concept of SCCSI to a system admitting multiple snapshot acquisition by rotating the dispersive element, so the dispersed spatio-spectral source is coded and integrated at different detector pixels in each rotation. Thus, a different set of coded projections is captured using the same optical components of the original architecture. The mathematical model of the multishot SCCSI system is presented along with several simulations. Results show that a gain up to 7 dB of peak signal-to-noise ratio is achieved when four SCCSI snapshots are compared to a single snapshot reconstruction. Furthermore, a gain up to 5 dB is obtained with respect to state-of-the-art architecture, the multishot CASSI.