By means of N-body+Hydrodynamics zoom-in simulations we study the evolution of the inner dark matter and stellar mass distributions of central dwarf galaxies formed in halos of virial masses \mbox { $M _ { v } $ } = 2 - 3 \times 10 ^ { 10 } h ^ { -1 } M _ { \odot } at z = 0 , both in a Warm Dark Matter ( WDM ) and Cold Dark Matter ( CDM ) cosmology . The half-mode mass in the WDM power spectrum of our simulations is \mbox { $M _ { f } $ } = 2 \times 10 ^ { 10 } h ^ { -1 } M _ { \odot } . In the dark matter ( DM ) only simulations halo density profiles are well described by the NFW parametric fit in both cosmologies , though the WDM halos have concentrations lower by factors of 1.5–2.0 than their CDM counterparts . In the hydrodynamic simulations , the effects of baryons significantly flatten the inner density , velocity dispersion , and pseudo phase-space density profiles of the WDM halos but not of the CDM ones . The density slope , measured at \approx 0.02 R _ { v } , \alpha _ { 0.02 } , becomes shallow in periods of 2 to 5 Gyr in the WDM runs . We explore whether this flattening process correlates with the global star formation ( SF ) , M _ { s } / M _ { v } ratio , gas outflow , and internal specific angular momentum histories . We do not find any clear trends , but when \alpha _ { 0.02 } is shallower than -0.5 , M _ { s } / M _ { v } is always between 0.25 % and 1 % . We conclude that the main reason of the formation of the shallow core is the presence of strong gas mass fluctuations inside the inner halo , which are a consequence of the feedback driven by a very bursty and sustained SF history in shallow gravitational potentials . Our WDM halos , which assemble late and are less concentrated than the CDM ones , obey these conditions . There are also ( rare ) CDM systems with extended mass assembly histories that obey these conditions and form shallow cores . The dynamical heating and expansion processes behind the DM core flattening apply also to the stars in a such a way that the stellar age and metallicity gradients of the dwarfs are softened , their stellar half-mass radii strongly grow with time , and their central surface densities decrease .