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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).