In this manuscript, corresponding to the weighted Ricci curvature, we discuss a class of Einstein Finsler metrics, which can be considered as a sequence of Einstein metrics on the general $CD(\infty,K)$ space. We investigate the non-Riemannian curvature conditions to ensure the equivalence of those metrics. With the bounds of the Ricci curvature, we can estimate the lower bounds of the S-curvature and the distortion. By arising a scalar curvature, we obtain an important formula relating the scalar curvature and the distortion. In some assumptions about some non-Riemannian curvatures, we get both the upper and lower bounds of the scalar curvature, the S-curvature and the distortion. We also discuss such bounds by using the lower bounds of the scalar curvature on the forward complete asymmetric $\infty$-Einstein Finsler manifold. The results in this manuscript will further lead us to study geometric analysis problems in general Finsler geometry.