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 ^ { 13 } CO 1 \to 0 , CS 2 \to 1 , and N _ { 2 } H ^ { + } 1 \to 0 survey of the Cygnus X region obtained with the FCRAO and CO , CS , HCO ^ { + } , N _ { 2 } H ^ { + } , and H _ { 2 } CO 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 ^ { 12 } CO are observed along and across the whole DR21 filament . By modelling the observed spectra , we obtain a typical infall speed of \sim 0.6 km s ^ { -1 } and mass accretion rates of the order of a few 10 ^ { -3 } M _ { \odot } yr ^ { -1 } for the two main clumps constituting the filament . These massive clumps ( 4900 and 3300 M _ { \odot } at densities of around 10 ^ { 5 } 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 can not 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 .