Nonequilibrium conditions fundamentally change how systems undergo phase separation. In lattice gases with attractive interactions under temperature gradients, particles have been shown to form periodic patterns and steady convective currents; however, a clear theoretical explanation of this convective phase separation is still missing. Here, we present a dynamical mean-field model that describes the mechanism behind this convective phase separation. Using linear stability analysis, we show that the transition from a uniform state to a periodic pattern is driven by the emergence of a dominant unstable mode. Numerical simulations confirm the predicted phase diagram and demonstrate that these convective currents are a robust feature of the steady state, appearing regardless of the initial conditions. These results provide a direct approach for understanding how temperature gradients drive the formation of steady-state convective patterns.