Context. Both theory and observations of star-forming clouds require simulations that combine the co-evolving chemistry, magneto-hydrodynamics, and radiative transfer in protostellar collapse simulation. A detailed knowledge of self-consistent chemical evolution for the main charge carriers (both gas species and dust grains) allows us to correctly estimate the rate and nature of magnetic dissipation in the collapsing core. This knowledge is critical to answer one of the most significant issues of star and planet formation: what is the magnitude and spatial distribution of magnetic flux as the initial condition to protoplanetary disk evolution?Aims. We use a chemo-dynamical version of RAMSES, which is described in a companion publication, to follow the chemo-dynamical evolution of collapsing dense cores with various dust properties and interpret differences that occur in magnetic diffusivity terms. These differences are crucial to circumstellar disk formation. Methods. We performed 3D chemo-dynamical simulations of 1 M⊙ isolated dense core collapse for a range in dust size assumptions. The number density of dust and its mean size affect the efficiency of charge capturing and the formation of ices. The radiative hydrodynamics and dynamical evolution of chemical abundances were used to reconstruct the magnetic diffusivity terms for clouds with various magnetisation. Results. The simulations are performed for a mean dust size ranging from 0.017 μm to 1 μm, and we adopt both a fixed dust size and a dust size distribution. The chemical abundances for this range of dust sizes are produced by RAMSES and serve as inputs to calculations of Ohmic, ambipolar, and Hall diffusivity terms. Ohmic resistivity only plays a role at the late stage of the collapse in the innermost region of the cloud where gas density is in excess of a few times 1013 cm-3. Ambipolar diffusion is a dominant magnetic diffusivity term in cases where mean dust size is a typical ISM value or larger. We demonstrate that the assumption of a fixed dominant ion mass can lead to a one order of magnitude mismatch in the ambipolar diffusion magnitude. The negative Hall effect is dominant during the collapse in case of small dust, i.e. for the mean dust size of 0.02 μm and smaller; we connect this effect to the dominance of negatively charged grains. We find that the Hall effect reverses its sign for mean dust size of 0.1 μm and smaller. The phenomenon of the sign reversal strongly depends on the number of negatively charged dust relative to the ions and the quality of coupling of the charged dust to the magnetic fields. We have adopted different strengths of magnetic fields, β = Pgas/Pmag = 2,5,25. We observe that the variation on the field strength only shifts the Hall effect reversal along the radius of the collapsing cloud, but does not prevent it. Conclusions. The dust grain mean size appears to be the parameter with the strongest impact on the magnitude of the magnetic diffusivity, dividing the collapsing clouds in Hall-dominated and ambipolar-dominated clouds and affecting the size of the resulting disks. We propose to link the dust properties and occurrence and size of disk structures in Class 0 young stellar objects. The proper accounting for dust grain growth in the radiative magneto-hydrodynamical collapse models are as important as coupling the dynamics of the collapse with the chemistry.