For simple graphs $X$ and $Y$ on $n$ vertices, the friends-and-strangers graph $\mathsf{FS}(X,Y)$ is the graph whose vertex set consists of all bijections $σ: V(X) \to V(Y)$, where two bijections $σ$ and $σ'$ are adjacent if and only if they agree on all but two adjacent vertices $a, b \in V(X)$ such that $σ(a), σ(b) \in V(Y)$ are adjacent in $Y$. Resolving a conjecture of Wang, Lu, and Chen, we completely characterize the connectedness of $\mathsf{FS}(X, Y)$ when $Y$ is a complete bipartite graph. We further extend this result to when $Y$ is a complete multipartite graph. We also determine when $\mathsf{FS}(X, Y)$ has exactly two connected components where $X$ is bipartite and $Y$ is a complete bipartite graph.