Let $f:X\to Y$ be a continuous map between topological spaces. Consider the quotient map $\pi:X\twoheadrightarrow X/E$ given by decomposing the fibers of $f$ to their connected components.
In Lemma 6.2.21 his book, Engelking proves that when $f:X\to Y$ is closed and has compact fibers, and $X$ is moreover Hausdorff, then $\pi$ is closed. The end of the proof makes crucial use of $f$ being closed.
On page 102 of the book Dynamic Topology by Whyburn & Duda, it is proved that if the connected components of the fibers of $f$ are compact, and $X,Y$ are Hausdorff with $X$ also locally compact, then $\pi$ is closed. To avoid assuming $f$ is closed, the corresponding part of the proof basically localizes to a compact subset on the base and then uses the fact any continuous map from a compact space to a Hausdorff space is closed.
It somehow feels possible to remove the assumptions on $X,Y$ and replace them by assumptions on $f$ without assuming $f$ is closed and still have $\pi$ be closed.
Definition. Say a continuous map $f:X\to Y$ is locally (universally) closed if for any open neighborhood $x\in W\subset X$, there exist neighborhoods $x\in U\subset W\subset X$ and $fy\in V\subset Y$ such that $fU\subset V$ and also the induced map $U\to V$ is (universally) closed.
Question. Suppose $f$ is separated and locally universally closed (suppose also compact fibers if necessary). Is $\pi$ closed?
The separation arguments in the proofs of Engelking and Whyburn seem to carry through when merely assuming $f$ is separated, so it really seems the only challenge is to remove the global assumption of $f$ being closed or the global assumption on the source and target spaces.