Determining the sensitivity of model outputs to input parameters is an important precursor to developing informative parameter studies, building surrogate models, and performing rigorous uncertainty quantification. A prominent class of models in many applications is dynamical systems whose trajectories lie on or near some attracting set after a sufficiently long time, and many quantities of interest are dominated by the systems’ dynamics on these attractors. Traditional methods of characterizing parameter sensitivities of time-varying quantities can be ill-suited to dynamics on an attractor by either overestimating discrepancies between trajectories on the same attractor or by averaging out important geometrical and dynamical features of the attractor and potentially losing critical information. We introduce a new sensitivity measure for such models by “lifting” the state space topology to a Wasserstein topology on a manifold of probability measures over the state space. Attractor dynamics are quantified by a unique invariant probability measure supported on the attractor, and parameter sensitivities are determined by computing the metric derivative of this family of measures in the Wasserstein topology. Nonintrusive approximation of invariant measures and Wasserstein derivatives, as well as applications to finite and infinite-dimensional problems, are discussed.