Learning-based methods for routing have gained significant attention in recent years, both in single-objective and multi-objective contexts. Y et, existing methods are unsuitable for routing on multigraphs, which feature multiple edges with distinct attributes between node pairs, despite their strong relevance in real-world scenarios. In this paper, we propose two graph neural network-based methods to address multi-objective routing on multigraphs. Our first approach operates directly on the multigraph by autoregressively selecting edges until a tour is completed. The second model, which is more scalable, first simplifies the multigraph via a learned pruning strategy and then performs autoregressive routing on the resulting simple graph. The field of neural combinatorial optimization has grown significantly in recent years and vehicle routing problems in particular have attracted much attention (Zhou et al., 2024a). While early works focused on the Traveling Salesman Problem (TSP) (Vinyals et al., 2015; Bello et al., 2017), new learning-based methods for vehicle routing can solve a wide range of problems efficiently and effectively, often surpassing classical state-of-the-art heuristics (Zhou et al., 2024b; Drakulic et al., 2025). Y et, existing methods have one limitation in common: they assume problems are defined on simple graphs. However, multigraph formulations, featuring several edges between each node pair, become relevant as soon as there are competing edges that cannot be chosen between a priori. Such situations typically occur when edges have more than one feature of interest, such as both travel time and distance. In spite of the high practical relevance of multigraph formulations (Lai et al., 2016; Ben Ticha et al., 2017), current learning-based methods are incapable of handling them due to two main reasons. Firstly, many state-of-the-art neural solvers rely on transformers to encode the problem instance. While these work well in the Euclidean setting (Kool et al., 2018) and with some modifications on asymmetric, directed graphs (Kwon et al., 2021), they lack the capability to encode multigraph structures. Secondly and more importantly, planning routes in multigraphs requires both selecting the node order and which edges to traverse, making current decoding strategies unsuitable. In this work, we aim to bridge the gap between learning-based methods for routing and accurate network representations given by multigraphs. We focus on the Multi-Objective (MO) setting, as several competing objectives naturally translates to several competing edges between each node pair. Nevertheless, our methods are general and can easily be extended to single-objective settings. Our code will be released publicly upon paper acceptance. To our knowledge, we present the first neural solvers designed for such structures.