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I think you need to replace $\dim f^{-1}(f(x))$ with $\dim_x f^{-1}(f(x))$ as in the linked question. I don't think the argument is using special global properties of affine varieties anywhere, so it should work the same way for non-affine varieties.
Sorry, I still don't see it. With this argument, the irreducibility hypothesis on $X$ would be unnecessary. The problem I see is that the Union of the sets you mentioned is not the set of all points such that $\dim f^{-1}(f(x)) \geq n$. There could be points in $W_i \setminus W_j$ whose $f_i$ fiber has dimension less than $n$, but which are in the same fiber as points in $W_j$ with $f_j$ fiber dimension greater than or equal to $n$: this is exactly the sort of thing that happens in the answers to this question and the linked one.
I'm afraid I'm missing something. Isn't is possible that if $Z$ is an irreducible component of $Y - U$, then $f^{-1}(Z)$ is not irreducible? In this case, I don't see how the recurrence goes through.
I know this is an old question, but I recently found it while reading about this problem. Be careful: in fact, the statement you give is not true even for $X$ irreducible when $X,Y$ are no longer supposed to be affine (and possibly when they are?) This answer gives a counterexample: mathoverflow.net/a/184925/56878
Thanks! In my case, I understand the derived pushforward reasonably well. The thing I don't understand is the edge map itself, especially on the torsion subgroup. I know the form of the sequence, but the Grothendieck formalism makes it hard to see what the ($E_2$) maps are. I want something like the CW-complex answer for fibrations. (If it helps: in my case, all fibers are $S^1$, but they have multiplicity at some points)
