Conditionally flat functors on spaces and groups

Consider a fibration sequence F → E → B of topological spaces which is preserved as such by some functor L, so that LF → LE → LB is again a fibration sequence. Pull the fibration back along an arbitrary map X → B into the base space. Does the pullback fibration enjoy the same property? For most functors this is not to be expected, and we concentrate mostly on homotopical localization functors. We prove that the only homotopical localization functors which behave well under pull-backs are nullifications. The same question makes sense in other categories. We are interested in groups and how localization functors behave with respect to group extensions. We prove that group theoretical nullification functors behave nicely, and so do all epireflections arising from a variety of groups.