We introduce a trajectory-based method for the inversion of flow and transport observations. The approach operates on the output of a standard numerical simulator and is applicable under very general conditions. Using only head and concentration histories, we efficiently construct the model parameter sensitivities required for solving the inverse problem. A single simulation or forward calculation is required in order to take a step in the inversion. Our formulation, based on asymptotic solutions for flow and transport, partitions the inverse problem into an arrival time and an amplitude-matching problem. Because the model parameter sensitivities are defined along trajectories, both inverse problems scale very well with respect to model size. At the migration experimental site of the Grimsel Rock Laboratory in Switzerland we utilized transient head and tracer arrival times to constrain the hydraulic conductivity between a suite of boreholes. The squared data misfit is reduced by over an order of magnitude in 30 iterations of our inversion algorithm. Each iteration requires two runs of our numerical simulator TOUGH2, one for each experiment, to construct the trajectories and associated model parameter sensitivities. We find a high-conductivity channel curving in a roughly east-west direction, in agreement with two previous studies. Estimates of model parameter resolution indicate that we can image the large-scale conductivity variations within the array of boreholes.