We present a simple stochastic, one-dimensional model for heat transfer in weakly collisional media as fusion plasmas. Energies of plasma particles are treated as lattice random variables interacting with a rate inversely proportional to their energy schematizing a screened Coulomb interaction. We consider both the equilibrium (microcanonical) and nonequilibrium case in which the system is in contact with heat baths at different temperatures. The model exhibits a characteristic length of thermalization that can be associated with an interaction mean free path and one observes a transition from ballistic to diffusive regime depending on the average energy of the system. A mean-field expression for heat flux is deduced from system heat transport properties. Finally, it is shown that the nonequilibrium steady state is characterized by long-range correlations. DOI: 10.1103/PhysRevE.87.023102