Envelope equations are derived for the convection rolls in the Rayleigh–Bénard–Poiseuille system, taking into account both their slow streamwise and transverse variations. At finite $O(1)$ Reynolds numbers, the stability of finite-amplitude longitudinal roll patterns is accessible to analysis in a moving frame of reference and stability is predicted provided a generalized Eckhaus criterion is satisfied. At lower Reynolds numbers, the analysis allows the analytical determination of the Green function for arbitrary orientations of the instability pattern. It clarifies previous results concerning the purely convective nature of all modes of instability except transverse rolls (for which a convective–absolute transition exists), as soon as the Reynolds number is non-zero.