The aim of this work is to build up a complete dynamical model of a molten carbonate fuel cell (MCFC) stack, describing both the thermo-fluid-dynamical and the electrochemical phenomena involved, i.e., both slow and (relatively) fast dynamics. Following a first-principle approach, a set of differential and algebraic equations is written, based on mass, momentum, energy, and charge balance referred to as small control volumes inside a cell. The outlined two-three-dimensional description takes into account the strong point-to-point anode and cathode reaction coupling due to gas crossflow. Simulations (carried out after suitable thermodynamical and electrochemical parameter tuning) highlight, for instance, the presence of dynamics, linked to the electrochemical behavior, with time constants on the order of a second; besides, rather fair matching to data which can be found in the literature is achieved, in terms of external potential difference and of electric power production. The obtained numerical results, therefore, support model correctness and reliability. This is useful in view of model-based cell operation analysis and control, both in stationary and in transient conditions.