Extremum-seeking control is a useful tool for the steady-state performance optimization of plants for which the dynamics and disturbance situation can be unknown. The case when steady-state plant outputs are constant received a lot of attention, however, in many applications time-varying outputs characterize plant performance. As a result, no static parameter-to-steady-state performance map can be obtained. In this work, an extremum-seeking control method is proposed that uses a so-called dynamic cost function to cope with these time-varying outputs. We show that, under appropriate conditions, the solutions of the extremum-seeking control scheme are uniformly ultimately bounded in view of bounded and time-varying external disturbances, and the region of convergence towards the optimal tunable plant parameters can be made arbitrarily small. Moreover, its working principle is illustrated by means of the performance optimal tuning of a variable-gain controller for a motion control application.