The effects of rainfall speed, V T , and meridional/vertical moisture gradients on the meridional moisture transport are examined in the context of mid-latitude baroclinic waves. These effects are investigated in an idealized model that can be solved analytically. The model is systematically derived in a precipitating quasi-geostrophic limit, starting from a moist atmospheric model with minimal representation of cloud microphysics. Single-phase dynamics are considered, with a comparison of three cases: unsaturated, saturated with V T = 0, and saturated with V T > 0. The Eady problem for linear baroclinic waves is analyzed, with modifications to incorporate moisture. As a preliminary step, the moist waves are shown to have properties consistent with prior studies, including larger growth rates and smaller spatial scales in the saturated cases in comparison to the classic dry Eady problem. Then, in addition, it is shown that the meridional moisture flux, as a function of height, has a mid-tropospheric maximum in the case of V T = 0, and a maximum in the lower troposphere or at the surface for sufficiently large values of V T . These results for different V T values are discussed in the context of meridional moisture transport in observational data.