Blue compact dwarf galaxies (BCDs) form stars at, for their sizes, extraordinarily high rates. In this paper, we study what triggers this starburst and what is the fate of the galaxy once its gas fuel is exhausted. We select four BCDs with smooth outer regions, indicating them as possible progenitors of dwarf elliptical galaxies. We have obtained photometric and spectroscopic data with the FORS and ISAAC instruments on the VLT. We analyse their infrared spectra using a full spectrum fitting technique, which yields the kinematics of their stars and ionized gas together with their stellar population characteristics. We find that the stellar velocity to velocity dispersion ratio ((nu/sigma)(star)) of our BCDs is of the order of 1.5, similar to...