Rate theories such as E-FGR have been widely used in the studies of polaritonic relaxation processes. In the present paper, we derived the analytic expression of the non-equilibrium FGR expression for the polaritonic relaxation process. A rigorous, discretization-independent, and time-domain derivation is provided for E-FGR based on the quantum master equation (QME), where NE-FGR arises as a natural generalization to E-FGR. This formulation of NE-FGR yields transfer rates between po-lariton states and dark states. In particular, for the |±⟩ → |D⟩ process, both the general formalism in Eq. 17 (with C ±→D (t, s) in Eq. 23) as well as the approximate expression k ±→D (t) (Eq. 29) are provided, which gives the analytic expression of the full time-dependent fluxside correlation function for the rate.