  According to star formation histories (SFHs), Local Group dwarf galaxies can
be broadly classified in two types: those forming most of their stars before
$z=2$ (${\it fast}$) and those with more extended SFHs (${\it slow}$). 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 ${\it fast}$ and ${\it slow}$ dwarfs, and
evaluate the impact of internal feedback by supernovae (SN) on the baryonic and
dark matter (DM) component of the dwarfs. ${\it Fast}$ dwarfs assembled more
stellar mass at early times and have larger amounts of DM within the half-light
radius than ${\it slow}$ dwarfs. By imposing that ${\it slow}$ 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 $\sim$10%. We find that internal
feedback alone appears insufficient to quench the SFH of ${\it fast}$ 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 ${\it fast}$ 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.
