  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.
