I believe it is well-known that for a based map $f:X\to Y$ of simplicial sets (possibly with some extra hypotheses on $X$ and $Y$), there is a long exact sequence $$ \ldots \to \pi_n(F)\to \pi_n(X) \to \pi_n(Y) \to \pi_{n-1}(F) \to \ldots, $$ where $F$ is the homotopy fibre of $f$. However, I cannot actually find a precise statement of this result. The closest I've come is this nLab page, which states essentially the same result for $\infty$-groupoids. I know that the classical model structure on simplicial sets is a presentation for the $(\infty,1)$-category of $\infty$-groupoids, so I suspect that maybe the result holds specifically for fibrant-cofibrant simplicial sets, but I'm not sure.
Ideally, I would like a reference that states the result explicitly for simplicial sets. If no such reference exists, I'd also be grateful for an explanation of the result.