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<i>Context. <i/>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. In addition, magnetic fields may play a decisive role in the star-formation process since they influence the efficiency of gas infall onto the protostar.<i>Aims. <i/>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.<i>Methods. <i/>We use molecular line data from our <sup>13<sup/>CO 1 0, CS 2 1, and N<sub>2<sub/>H<sup>+<sup/> 1 0 survey of the Cygnus X region obtained with the FCRAO and high-angular resolution observations in isotopomeric lines of CO, CS, HCO<sup>+<sup/>, N<sub>2<sub/>H<sup>+<sup/>, and H<sub>2<sub/>CO, obtained with the IRAM 30 m telescope, to investigate the distribution of the different phases of molecular gas. Gravitational infall is identified by the presence of inverse P Cygni profiles that are detected in optically thick lines, while the optically thinner isotopomers are found to reach a peak in the self-absorption gap.<i>Results. <i/>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<sup>+<sup/> and <sup>12<sup/>CO are observed along and across the whole DR21 filament. By modelling the observed spectra, we obtain a typical infall speed of ~0.6 km s<sup>-1<sup/> and mass accretion rates of the order of a few 10<sup>-3<sup/> yr<sup>-1<sup/> for the two main clumps constituting the filament. These massive clumps (4900 and 3300 at densities of around 10<sup>5<sup/> cm<sup>-3<sup/> within 1 pc diameter) are both gravitationally contracting (with free-fall times much shorter than sound crossing times and low virial parameter <i>α<i/>). 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.<i>Conclusions. <i/>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.