The authors propose a new proof of the strong displacement convexity for a class of integral functionals defined over a compact Riemannian manifold satisfying a lower Ricci curvature bound. The approach adopted does not rely on existence and regularity results for optimal transport maps on Riemannian manifolds, but it is based on the Eulerian point of view recently introduced by F. Otto and M. Westdickenberg [SIAM J. Math. Anal. 37, No. 4, 1227–1255 (2006; Zbl 1094.58016)] and on the metric characterization of the gradient flows generated by the functionals in the Wasserstein space.