We describe the application of tools from dynamical systems to define and quantify the unsteady fluid transport that occurs during fluid–structure interactions and in unsteady recirculating flows. The properties of Lagrangian coherent structures (LCS) are used to enable analysis of flows with arbitrary time-dependence, thereby extending previous analytical results for steady and time-periodic flows. The LCS kinematics are used to formulate a unique, physically motivated definition for fluid exchange surfaces and transport lobes in the flow. The methods are applied to numerical simulations of two-dimensional flow past a circular cylinder at a Reynolds number of 200; and to measurements of a freely swimming organism, the Aurelia aurita jellyfish. The former flow provides a canonical system in which to compare the present geometrical analysis with classical, Eulerian (e.g. vortex shedding) perspectives of fluid–structure interactions. The latter flow is used to deduce the physical coupling that exists between mass and momentum transport during self-propulsion. In both cases, the present methods reveal a well-defined, unsteady recirculation zone that is not apparent in the corresponding velocity or vorticity fields. Transport rates between the ambient flow and the recirculation zone are computed for both flows. Comparison of fluid transport geometry for the cylinder crossflow and the self-propelled swimmer within the context of existing theory for two-dimensional lobe dynamics enables qualitative localization of flow three-dimensionality based on the planar measurements. Benefits and limitations of the implemented methods are discussed, and some potential applications for flow control, unsteady propulsion, and biological fluid dynamics are proposed.