This paper adopts a detailed 3D mesoscale modelling strategy to tackle the pressing challenge of assessment for masonry arch bridges. Most of these structures have undergone severe deterioration during their service-life under both increasing traffic loads and environmental actions. However, they still represent a major portion of existing infrastructure for roadway and railway traffic in Europe. The development of reliable assessment methods for such structures is urgently required. However, this is clearly hindered by their complex behaviour. In the adopted modelling strategy, the heterogeneous nature of masonry arch bridges is addressed by adopting a discrete mesoscale modelling for masonry components in conjunction with a continuous modelling for backfill. In addition, the interaction between the different bridge parts is accounted for using nonlinear interface elements at the physical interfaces, where a mesh tying approach is employed to connect non-conforming meshes. The computational effort associated with such detailed models is efficiently optimised using a domain partitioning technique, where the bridge is subdivided into smaller partitions allowing for an efficient parallel computation. This modelling strategy aims at capturing the overall behaviour of masonry arch bridges, which is dominantly three-dimensional mainly due to the asymmetry of typical traffic loads and the complex geometry in the case of skew bridges. In the paper, the response of realistic 3D bridge samples with various loading and geometric configurations is investigated leading to an improved understanding of their typical 3D behaviour to collapse.