Context. The formation of massive stars is a highly complex process in which it is unclear whether the star-forming gas is in global gravitational collapse or an equilibrium state supported by turbulence and/or magnetic fields. Aims. By studying one of the most massive and dense star-forming regions in the Galaxy at a distance of less than 3 kpc, i.e. the filament containing the well-known sources DR21 and DR21(OH), we attempt to obtain observational evidence to help us to discriminate between these two views. Methods. We use molecular line data from our 13CO 1->0, CS 2->1, and N2H+ 1->0 survey of the Cygnus X region obtained with the FCRAO and CO, CS, HCO+, N2H+, and H2CO data obtained with the IRAM 30m telescope. Results. We observe a complex velocity field and velocity dispersion in the DR21 filament in which regions of the highest column density, i.e., dense cores, have a lower velocity dispersion than the surrounding gas and velocity gradients that are not (only) due to rotation. Infall signatures in optically thick line profiles of HCO+ and 12CO are observed along and across the whole DR21 filament. By modeling the observed spectra, we obtain a typical infall speed of ~0.6 km s-1 and mass accretion rates of the order of a few 10?3 M? yr-1 for the two main clumps constituting the filament. These massive clumps (4900 and 3300 M? at densities of around 105 cm?3 within 1 pc diameter) are both gravitationally contracting. The more massive of the clumps, DR21(OH), is connected to a sub-filament, apparently 'falling' onto the clump. This filament runs parallel to the magnetic field. Conclusions. All observed kinematic features in the DR21 filament (velocity field, velocity dispersion, and infall), its filamentary morphology, and the existence of (a) sub-filament(s) can be explained if the DR21 filament was formed by the convergence of flows on large scales and is now in a state of global gravitational collapse. Whether this convergence of flows originated from self-gravity on larger scales or from other processes cannot be determined by the present study. The observed velocity field and velocity dispersion are consistent with results from (magneto)-hydrodynamic simulations where the cores lie at the stagnation points of convergent turbulent flows. Show Partial Abstract