We show that many classical optimization problems --- such as $(1\pmε)$-approximate maximum flow, shortest path, and transshipment --- can be computed in $\newcommand{\tmix}{τ_{\text{mix}}}\tmix(G)\cdot n^{o(1)}$ rounds of distributed message passing, where $\tmix(G)$ is the mixing time of the network graph $G$. This extends the result of Ghaffari et al.\ [PODC'17], whose main result is a distributed MST algorithm in $\tmix(G)\cdot 2^{O(\sqrt{\log n \log\log n})}$ rounds in the CONGEST model, to a much wider class of optimization problems. For many practical networks of interest, e.g., peer-to-peer or overlay network structures, the mixing time $\tmix(G)$ is small, e.g., polylogarithmic. On these networks, our algorithms bypass the $\tildeΩ(\sqrt n+D)$ lower bound of Das Sarma et al.\ [STOC'11], which applies for worst-case graphs and applies to all of the above optimization problems. For all of the problems except MST, this is the first distributed algorithm which takes $o(\sqrt n)$ rounds on a (nontrivial) restricted class of network graphs. Towards deriving these improved distributed algorithms, our main contribution is a general transformation that simulates any work-efficient PRAM algorithm running in $T$ parallel rounds via a distributed algorithm running in $T\cdot \tmix(G)\cdot 2^{O(\sqrt{\log n})}$ rounds. Work- and time-efficient parallel algorithms for all of the aforementioned problems follow by combining the work of Sherman [FOCS'13, SODA'17] and Peng and Spielman [STOC'14]. Thus, simulating these parallel algorithms using our transformation framework produces the desired distributed algorithms. The core technical component of our transformation is the algorithmic problem of solving \emph{multi-commodity routing}---that is, roughly, routing $n$ packets each from a given source to a given destination---in random graphs. For this problem, we obtain a...