Liquid crystals have been very fruitful systems to study equilibrium phase transitions. Recently, they have become an important system to study dynamics of first-order phase transitions. The moving nonequilibrium nematic-isotropic interface is a model system to study growth of stable states into metastable states and displays a myriad of dynamical instabilities that, far from equilibrium, drive the system to a scenario of spatio-temporal chaos. We present a mean-field theory for the time evolution of a planar nonequilibrium nematic-isotropic interface for pure liquid crystals using a time dependent Ginzburg-Landau equation, which is one of the simplest approaches to dissipative dynamics. We obtain a theoretical expression for the growth kinetics of the nematic phase into a metastable isotropic phase and compare it with our experimental results. In a directional solidification arrangement we study instabilities of the nematic-isotropic interface of the liquid crystal 8CB doped with water and hexachloroethane. The observed instabilities are similar to cellular instabilities that appear during growth of crystal-melt interfaces of binary mixtures. We then compare our results with known theories of morphological instabilities during crystal growth.