Cells are active systems with molecular force generation that drives complex dynamics at the supramolecular scale. We present a quantitative study of molecular motions in cells over times from milliseconds to hours. Noninvasive tracking was accomplished by imaging highly stable near-infrared luminescence of single-walled carbon nanotubes targeted to kinesin-1 motor proteins in COS-7 cells. We observed a regime of active random "stirring" that constitutes an intermediate mode of transport, different from both thermal diffusion and directed motor activity. High-frequency motion was found to be thermally driven. At times greater than 100 milliseconds, nonequilibrium dynamics dominated. In addition to directed transport along microtubules, we observed strong random dynamics driven by myosins that result in enhanced nonspecific transport. We present a quantitative model connecting molecular mechanisms to mesoscopic fluctuations.