We present a closed form for the governing equations of chemically reacting flows under local thermodynamic equilibrium, which rigorously takes into account effects of elemental (de)mixing. To this end, we show that when chemistry is fast, the diffusion fluxes of elements and species enthalpies can be expressed as explicit linear functions of gradients of elemental mass fractions and temperature. Our formulation is a natural extension of classical work on local equilibrium flows by other authors and yields results equivalent with a recent fully rigorous mathematical theory in a straightforward and physically appealing manner. The obtained set of equations is well-suited for numerical implementations and does not require the computationally expensive evaluation of thermodynamic derivatives using finite differences. The new transport coefficients that appear in the equations allow quantitative predictions and help to gain deeper insight into the physics of chemically reacting flows at and near local equilibrium.