We introduce a perturbation-based extremum-seeking controller for general nonlinear dynamical plants with an arbitrary number of tunable plant parameters. The controller ensures asymptotic convergence of the plant parameters to their performance-optimizing values for any initial plant condition under the assumptions in this work. The key to this result is that the amplitude and the frequencies of the perturbations, as well as other tuning parameters of the controller, are time varying. Remarkably, the time-varying tuning parameters can be chosen such that asymptotic convergence is achieved for all plants that satisfy the assumptions, thereby guaranteeing stability of the resulting closed-loop system of plant and controller regardless of tuning.