Thermonuclear burn criteria, with charged-particle energy deposition, in fusion plasmas using a perturbative expansion of the coupled burn and transport equations about any quasi-equilibrium temperature are examined. Burn propagation and energy deposition are coupled in a reaction wave model, and effects are quantified using linearized one-temperature-plus-diffusion equations. Eigenvalue growth rate and propagation criteria, which depend on plasma properties and alpha mean-free-paths, are described. Effective cross sections appropriate to random mixtures are discussed, and burn propagation and energy deposition in limiting cases of homogeneous and heterogeneous media are contrasted. Methodology is general to thermonuclear processes, but our focus is deuterium-tritium burn in the reaction d + t → n + α.