  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 infra-red 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 (v/sigma) of our BCDs is of the order of 1.5, similar to that
of dwarf elliptical galaxies. Thus, those objects do not require significant
(if any) loss of angular momentum to fade into early type dwarfs. This finding
is in discordance with previous studies, which however compared the stellar
kinematics of dwarf elliptical galaxies with the gaseous kinematics of star
forming dwarfs. The stellar velocity fields of our objects are very disturbed
and the star-formation regions are often kinematically decoupled from the rest
of the galaxy. These regions can be more or less metal rich with respect to the
galactic body, and sometimes they are long lived. These characteristics prevent
us from pinpointing a unique trigger of the star formation, even within the
same galaxy. Gas impacts, mergers, and in-spiraling gas clumps are all possible
star-formation ignitors for our targets.
