We introduce action-driven flows for causal variational principles, being a class of nonconvex variational problems emanating from applications in fundamental physics. In the compact setting, Hölder continuous curves of measures are constructed by using the method of minimizing movements. As is illustrated in examples, these curves will in general not have a limit point, due to the nonconvexity of the action. This leads us to introducing a novel penalization which ensures the existence of a limit point, giving rise to approximative solutions of the Euler-Lagrange equations. The methods and results are adapted and generalized to the causal action principle in the finite-dimensional case. As an application, we construct a flow of measures for causal fermion systems in the infinite-dimensional situation.