We present a novel global-in-time variational approach to gradient flows and doubly nonlinear equations in (reflexive) Banach spaces. It is based on the De Giorgi's principle, which states that solving a gradient flow is equivalent to being a null-minimizer of a suitable energy functional among all trajectories sharing the same initial position. As for the similar Brezis-Ekeland-Nayroles (BEN) principle (which applies only to a convex framework), finding a minimizer for such functional is not difficult in general, but proving that the minimum is zero poses a real challenge. In the BEN formulation, the task has been accomplished by Ghoussoub, resorting to the tool of self-dual Lagrangians. Our approach allows to extend the analysis to nonconvex energies, directly dealing with the De Giorgi's functional, and it relies on a convexification of the problem in spaces of measures exploiting the so-called superposition principle. The validity of the null-minimization is then recovered by a careful application of the Von Neumann minimax theorem, and by employing the "backward boundedness" property of the dual Hamilton-Jacobi equation. The talk is based on a joint work with A. Pinzi and G. Savaré.