According to star formation histories (SFHs), Local Group dwarf galaxies can be broadly classified in two types: those forming most of their stars before z = 2 (fast) and those with more extended SFHs (slow). The most precise SFHs are usually derived from deep but not very spatially extended photometric data; this might alter the ratio of old to young stars when age gradients are present. Here, we correct for this effect and derive the mass formed in stars by z = 2 for a sample of 16 Local Group dwarf galaxies. We explore early differences between fast and slow dwarfs, and evaluate the impact of internal feedback by supernovae (SNe) on the baryonic and dark matter (DM) component of the dwarfs. Fast dwarfs assembled more stellar mass at early times and have larger amounts of DM within the half-light radius than slow dwarfs. By imposing that slow dwarfs cannot have lost their gas by z = 2, we constrain the maximum coupling efficiency of SN feedback to the gas and to the DM to be ˜10 per cent. We find that internal feedback alone appears insufficient to quench the SFH of fast dwarfs by gas deprivation, in particular for the fainter systems. Nonetheless, SN feedback can core the DM halo density profiles relatively easily, producing cores of the sizes of the half-light radius in fast dwarfs by z = 2 with very low efficiencies. Amongst the `classical' Milky Way satellites, we predict that the smallest cores should be found in Draco and Ursa Minor, while Sculptor and Fornax should host the largest ones.