We prove the upper and lower bounds of the diameter of a compact manifold (M , g (t)) with dim R M = n (n ≥ 3) and a family of Riemannian metrics g (t) satisfying some geometric flows. Except for Ricci flow, these flows include List–Ricci flow, harmonic-Ricci flow, and Lorentzian mean curvature flow on an ambient Lorentzian manifold with non-negative sectional curvature.