The formation of massive stars is a highly complex process in which it is not clear whether the star-forming gas is in global gravitational collapse or in an equilibrium state, supported by turbulence. By studying one of the most massive and dense star-forming regions in the Galaxy at a distance of less than 3 kpc, the filament containing the well-known sources DR21 and DR21(OH), we expect to find observational signatures that allow to discriminate between the two views. 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 high-angular resolution observations of CO, CS, HCO+, N2H+, and H2CO, obtained with the IRAM 30m telescope. We observe a complex velocity field and velocity dispersion in the DR21 filament in which regions of 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. From modelling the observed spectra, we obtain a typical infall speed of 0.6 km/s and mass accretion rates of the order of a few 10^-3 Msun/yr for the two main clumps constituting the filament. These massive (4900 and 3300 Msun) clumps are both gravitationally contracting. All observed kinematic features in the DR21 filament can be explained if it is formed by the convergence of flows at large scales and is now in a state of global gravitational collapse. Whether this convergence of flows originated from self-gravity at larger scales or from other processes can not be settled with 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.Read less <