Cellular Potts models are broadly applied across developmental biology and cancer research. We overcome limitations of the traditional approach, which reinterprets a modified Metropolis sampling as ad hoc dynamics, by introducing a physical timescale through Poissonian kinetics and by applying principles of stochastic thermodynamics to separate thermal and relaxation effects from athermal noise and nonconservative forces. Our method accurately describes cell-sorting dynamics in mouse-embryo development and identifies the distinct contributions of nonequilibrium processes, e.g., cell growth and active fluctuations.