In the context of the Local Cosmology from Isolated Dwarfs (LCID) project, we present detailed star formation and chemical enrichment histories for six isolated Local Group dwarf galaxies. One of the main goals of the LCID project is to investigate the influence of global processes like cosmic reionization, or internal mechanisms such as SNe feedback, on the evolution of dwarf galaxies. We show that all galaxies in the sample, including those that ended star formation very early on, such as the dSph and dIrr/dSph types, formed most of their stars after reionization was complete. This demonstrates that reionization alone was not able to stop star formation in them, as had been expected. We explore alternative mechanisms that may have contributed to the early stop of star formation in these galaxies.