We propose a new dynamic algorithm which transports samples from a reference distribution to a target distribution in unit time, given access to the target-to-reference density ratio. Our approach is to seek a sequence of transport maps that push forward the reference along a path given by a geometric mixture of the two densities. We take the maps to be simply parameterized, local, sample-driven optimal transport maps which we identify by approximately solving a root-finding problem formulated using importance weights. When feature functions for the maps are taken to be kernels, we obtain a novel interacting particle system from which we derive finite-particle and mean-field ODEs. In discrete time, we introduce an adaptive algorithm for simulating this interacting particle system which adjusts the ODE time steps based on the quality of the transport, automatically uncovering a good "schedule" for traversing the geometric mixture of densities.