We study the deconfinement of hadronic matter into quark matter in a protoneutron star focusing on the effects of the finite size on the formation of just-deconfined color superconducting quark droplets embedded in the hadronic environment. The hadronic phase is modeled by the non-linear Walecka model at finite temperature including the baryon octet and neutrino trapping. For quark matter we use an $SU(3)_f$ Nambu-Jona-Lasinio model including color superconductivity. The finite size effects on the just deconfined droplets are considered in the frame of the multiple reflection expansion. In addition, we consider that just deconfined quark matter is transitorily out of equilibrium respect to weak interaction, and we impose color neutrality and flavor conservation during the transition. We calculate self-consistently the surface tension and curvature energy density of the quark hadron inter-phase and find that it is larger than the values typically assumed in the literature. The transition density is calculated for drops of different sizes, and at different temperatures and neutrino trapping conditions. Then, we show that energy-density fluctuations are much more relevant for deconfinement than temperature and neutrino density fluctuations. We calculate the critical size spectrum of energy-density fluctuations that allows deconfinement as well as the nucleation rate of each critical bubble. We find that drops with any radii smaller than 800 fm can be formed at a huge rate when matter achieves the bulk transition limit of 5-6 times the nuclear saturation density.
