In this talk, we review comparison theorems under variable lower weighted Ricci curvature bounds with so-called $\epsilon$-range. This notion applies to both weighted Riemannian and Lorentzian manifolds (as well as Finsler and Lorentz-Finsler manifolds), and was inspired by Wylie-Yeroshkin's comparison theorems for weighted Riemannian manifolds of 1-Ricci curvature bounded below by using the weight function. By introducing the $\epsilon$-range, we can unify the usual Ricci curvature bounds by constants and Wylie-Yeroshkin's variable bounds into a single framework. This is joint work with Yufeng Lu and Ettore Minguzzi.