We describe a version of the FGLM algorithm that can be used to compute generic fibers of positive-dimensional polynomial ideals. It combines the FGLM algorithm with a Hensel lifting strategy. We show that this algorithm has a complexity quasi-linear in the number of lifting steps. Some provided experimental data also demonstrates the practical efficacy of our algorithm. Additionally, we sketch a related Hensel lifting method to compute Gröbner bases using so-called tracers.