Abstract To properly quantify the possible residual systematic errors affecting the classical Cepheid distance scale, a detailed theoretical scenario is recommended. By extending the set of nonlinear, convective pulsation models published for Z = 0.02 to Z = 0.004, Z = 0.008, and Z = 0.03, we provide a detailed homogeneous, nonlinear model grid taking into account simultaneous variations of the mass–luminosity relation, the efficiency of superadiabatic convection, and the chemical composition. The dependence of the inferred period–radius, period–mass–radius, and period–mass–luminosity–temperature relations on the input parameters is discussed for both the fundamental and first overtone modes. The trend of the instability strip getting redder as the metallicity increases is confirmed for the additional mass–luminosity assumptions and mixing length values. From the obtained multifilter light curves, we derive the mean magnitudes and colors, and in turn the period–luminosity–color and period–Wesenheit relations, for each assumed chemical composition, mass–luminosity relation, and efficiency of superadiabatic convection. Application to a well-studied sample of Cepheids in the Large Magellanic Cloud allows us to constrain the dependence of the inferred distance modulus on the assumed mass–luminosity relation, and the inclusion of the metallicity term in the derivation of the period–Wesenheit relations allows us, for each assumed mass–luminosity relation, to predict the metallicity dependence of the Cepheid distance scale. The obtained metal-dependent, period–Wesenheit relations are compared with recent results in the literature and applied to a sample of Gaia Early Data Release 3 Galactic Cepheids with known metal abundances to derive individual parallaxes. The comparison of these predictions with Gaia results is finally discussed.