Four-dimensional phase computed tomography (4D phase CT) by an X-ray Talbot interferometer (XTI) with white synchrotron radiation has ever been demonstrated at a temporal resolution of about 1 s for soft-matter samples. However, the radiation damage to samples caused by white synchrotron radiation occasionally hampers our understanding of the sample dynamical properties. Based on the fact that XTI functions with X-rays of a bandwidth up to ca. 10% with performance comparable to that by monochromatic X-rays, filtering white synchrotron radiation to generate a 'pink-beam' of a 10% bandwidth is effective to reduce radiation damage without degrading the image quality and temporal resolution. We have therefore developed pink-beam 4D phase CT at SPring-8, Japan by installing a multilayer mirror with a 10% bandwidth and a 25 keV central photon energy. XTI optimal at this photon energy was built downstream, and a CMOS-based X-ray detector was used to achieve fast image acquisitions with an exposure time of 1 ms (or 0.5 ms) per moiré image. The resultant temporal resolution of pink-beam 4D phase CT was 2 s (1 s). We applied the pink-beam 4D phase CT to in-situ observation of polypropylene, poly(methyl methacrylate), and polycarbonate under infrared laser irradiation (1064 nm). The dynamics of melting, bubbling, and ashing were successfully visualized in 3D movies without problematic radiation damage by synchrotron radiation.