We study the hydrodynamical transition from an hadronic star into a quark or a hybrid star. We discuss the possible mode of burning, using a fully relativistic formalism and realistic Equations of State in which hyperons can be present. We take into account the possibility that quarks form a diquark condensate. We also discuss the formation of a mixed phase of hadrons and quarks, and we indicate which region of the star can rapidly convert in various possible scenarios. An estimate of the final temperature of the system is provided. We find that the conversion process always corresponds to a deflagration and never to a detonation. Hydrodynamical instabilities can develop on the front. We estimate the increase in the conversion's velocity due to the formation of wrinkles and we find that, although the increase is significant, it is not sufficient to transform the deflagration into a detonation in essentially all realistic scenarios. Concerning convection, it does not always develop. In particular the system does not develop convection if hyperons are not present in the initial phase and if the newly formed quark phase is made of ungapped (or weakly gapped) quarks. At the contrary, the process of conversion from ungapped quark matter to gapped quarks always allows the formation of a convective layer. Finally, we discuss possible astrophysical implications of our results.

47 pages, 23 figures New version after referee's comments. Two sections and two figures added