Let $E\longrightarrow X$ be a surface (with holes) bundle. The structure group is then $M_{g, s}$, the mapping class group of the fiber. It follows from the famous work of Penner that the classifying space of $M_{g, s}$ is homotopy equivalent to the geometrical realization of the category of fatgraphs, i.e., $$BM_{g,s} \simeq |\mathit{Fat}_{g,s}|.$$ My question is as follows: how can one construct the classifying map $X\longrightarrow |\mathit{Fat}_{g,s}|$ explicitly given the initial bundle in these terms? This seems pretty intuitive, but I can not get to the formal description.

My only thought was to try to find something similar the natural map to the Milnor join construction of the classifying space.

Edit: the natural restrictions for the genus and number of holes are $s\geqslant 1$ and $s+2g \geqslant 3$.