Context. Small dust grains are essential ingredients of star, disk and planet formation.Aims. We present an Eulerian numerical approach to study small dust grain dynamics in the context of star and protoplanetary disk formation. It is designed for finite volume codes. We use it to investigate dust dynamics during the protostellar collapse.Methods. We present a method to solve the monofluid equations of gas and dust mixtures with several dust species in the diffusion approximation implemented in the adaptive-mesh-refinement code RAMSES. It uses a finite volume second-order Godunov method with a predictor-corrector MUSCL scheme to estimate the fluxes between the grid cells.Results. We benchmark our method against six distinct tests, DUSTYADVECT, DUSTYDIFFUSE, DUSTYSHOCK, DUSTYWAVE, SETTLING, and DUSTYCOLLAPSE. We show that the scheme is second-order accurate in space on uniform grids and intermediate between second- and first-order on non-uniform grids. We apply our method on various DUSTYCOLLAPSE simulations of 1 M⊙ cores composed of gas and dust.Conclusions. We developed an efficient approach to treat gas and dust dynamics in the diffusion regime on grid-based codes. The canonical tests were successfully passed. In the context of protostellar collapse, we show that dust is less coupled to the gas in the outer regions of the collapse where grains larger than ≃100 μm fall significantly faster than the gas.