Geometrical forms of products are important quality characteristics in advanced manufacturing, e.g., geometries of 3D printed products and morphology of nano-manufactured products. In many material-processing systems, particle populations are heterogeneous, multi-modal and their shape distributions may evolve over time due to uncertain physical and environmental conditions. Hence, tracking the temporal behavior of shape distributions is important for process monitoring, material characterization, and data-driven decision-making. Here, we propose a dynamic mixture-modeling framework for time-varying shape distributions, where particle shapes are represented through clusters and the corresponding mixing proportions are modeled as time-dependent quantities. In this study, we consider that only mixture proportions will change along with time; the other parameters like mean shape and covariance will be constant. Finally, the proposed approach is evaluated using nanoparticle shape-distribution data observed over 20 timepoints.