What you're looking for is Quillen's Theorem B. Roughly speaking, it says that if all the "combinatorial homotopy fibers" of f are homotopy equivalent (via base change along morphisms in the poset), then the map is a fibration up to homotopy (meaning the inclusion of the ordinary fiber into the homotopy fiber is a homotopy equivalence). It originally appeared in Higher Algebraic K-theory, I (Lecture Notes in Math 341).

What you're looking for might be Quillen's Theorem B. Roughly speaking, it says that if all the "combinatorial homotopy fibers" are homotopy equivalent (via base change along morphisms in the poset), then the combinatorial homotopy fibers are weakly equivalent to the honest homotopy fibers. It originally appeared in Higher Algebraic K-theory, I (Lecture Notes in Math 341).