This paper proposes two graph neural network-based methods for solving routing problems in multi-objective multi-graphs. The first method autorecursively selects edges from the multi-graph to complete a path. The second, more scalable method, simplifies the multi-graph using a learned pruning strategy and then performs autorecursive routing on the simplified graph. Experimentally evaluating both models across a variety of problems and graph distributions, we demonstrate competitive performance compared to robust heuristics and neural network-based baseline models. Our goal is to overcome the limitations of existing single-objective or multi-objective routing methods in adapting to multi-graphs.