Context. Dust plays a key role during star, disk, and planet formation. Yet, its dynamics during the protostellar collapse remain a poorly investigated field. Recent studies seem to indicate that dust may decouple efficiently from the gas during these early stages.Aims. We aim to understand how much and in which regions dust grains concentrate during the early phases of the protostellar collapse, and to see how this depends on the properties of the initial cloud and of the solid particles.Methods. We used the multiple species dust dynamics MULTIGRAIN solver of the grid-based code RAMSES to perform various simulations of dusty collapses. We performed hydrodynamical and magnetohydrodynamical simulations where we varied the maximum size of the dust distribution, the thermal-to-gravitational energy ratio, and the magnetic properties of the cloud. We simulated the simultaneous evolution of ten neutral dust grain species with grain sizes varying from a few nanometers to a few hundreds of microns.Results. We obtain a significant decoupling between the gas and the dust for grains of typical sizes of a few tens of microns. This decoupling strongly depends on the thermal-to-gravitational energy ratio, the grain sizes, and the inclusion of a magnetic field. With a semi-analytic model calibrated on our results, we show that the dust ratio mostly varies exponentially with the initial Stokes number at a rate that depends on the local cloud properties.Conclusions. We find that larger grains tend to settle and drift efficiently in the first-core and in the newly formed disk. This can produce dust-to-gas ratios of several times the initial value. Dust concentrates in high-density regions (cores, disk, and pseudo-disk) and is depleted in low-density regions (envelope and outflows). The size at which grains decouple from the gas depends on the initial properties of the clouds. Since dust cannot necessarily be used as a proxy for gas during the collapse, we emphasize the necessity of including the treatment of its dynamics in protostellar collapse simulations.