Context. To form metre-sized pre-planetesimals in protoplanetary discs, growing grains have to decouple from the gas before they are accreted onto the central star during their phase of fast radial migration and thus overcome the so-called 'radial-drift barrier' (often inaccurately referred to as the 'metre-size barrier'). Aims. We predict the outcome of the radial motion of dust grains in protoplanetary discs whose surface density and temperature follow power-law profiles, with exponent p and q respectively. We investigate both the Epstein and the Stokes drag regimes which govern the motion of the dust. Methods. We analytically integrate the equations of motion obtained from perturbation analysis. We compare these results with those from direct numerical integration of the equations of motion. Then, using data from observed discs, we predict the fate of dust grains in real discs. Results. When a dust grain reaches the inner regions of the disc, the acceleration due to the increase of the pressure gradient is counterbalanced by the increase of the gas drag. We find that most grains in the Epstein (resp. the Stokes) regime survive their radial migration if-p + q + 1/2 ≤ 0 (resp. if q ≤ 2/3). The majority of observed discs satisfies both-p + q + 1/2 ≤ 0 and q ≤ 2/3: a large fraction of both their small and large grains remain in the disc, for them the radial drift barrier does not exist.