We prove that the transport of any differentiable scalar observable in $d$-dimensional non-equilibrium systems is bounded from above by the total entropy production scaled by the amount the observation "stretches" microscopic coordinates. The result--a time-integrated generalized speed limit--reflects the thermodynamic cost of transport of observables, and places underdamped and overdamped stochastic dynamics on equal footing with deterministic motion. Our work allows for stochastic thermodynamics to make contact with bulk experiments, and fills an important gap in thermodynamic inference, since microscopic dynamics is, at least for short times, underdamped. Requiring only averages but not sample-to-sample fluctuations, the proven transport bound is practical and applicable not only to single-molecule but also bulk experiments where only averages are observed, which we demonstrate by examples. Our results may facilitate thermodynamic inference on molecular machines without an obvious directionality from bulk observations of transients probed, e.g.\ in time-resolved X-ray scattering.