We report high-quality Hα/CO imaging spectroscopy of nine massive (log median stellar mass = 10.65 M ⊙) disk galaxies on the star-forming main sequence (henceforth SFGs), near the peak of cosmic galaxy evolution (z ∼ 1.1-2.5), taken with the ESO Very Large Telescope, IRAM-NOEMA, and Atacama Large Millimeter/submillimeter Array. We fit the major axis position-velocity cuts with beam-convolved, forward models with a bulge, a turbulent rotating disk, and a dark matter (DM) halo. We include priors for stellar and molecular gas masses, optical light effective radii and inclinations, and DM masses from our previous rotation curve analysis of these galaxies. We then subtract the inferred 2D model-galaxy velocity and velocity dispersion maps from those of the observed galaxies. We investigate whether the residual velocity and velocity dispersion maps show indications for radial flows. We also carry out kinemetry, a model-independent tool for detecting radial flows. We find that all nine galaxies exhibit significant nontangential flows. In six SFGs, the inflow velocities (v r ∼ 30-90 km s−1, 10%-30% of the rotational component) are along the minor axis of these galaxies. In two cases the inflow appears to be off the minor axis. The magnitudes of the radial motions are in broad agreement with the expectations from analytic models of gravitationally unstable, gas-rich disks. Gravitational torques due to clump and bar formation, or spiral arms, drive gas rapidly inward and result in the formation of central disks and large bulges. If this interpretation is correct, our observations imply that gas is transported into the central regions on ∼10 dynamical timescales.