The response up to collapse of masonry arch bridges is very complex and affected by many uncertainties. In general, accurate response predictions can be achieved using sophisticated numerical descriptions, requiring a significant number of parameters that need to be properly characterised. This study focuses on the sensitivity of the behaviour of masonry arch bridges with respect to a wide range of mechanical parameters considered within a detailed modelling approach. The aim is to investigate the effect of constitutive parameters variations on the stiffness and ultimate load capacity under vertical loading. First, advanced numerical models of masonry arches and of a masonry arch bridge are developed, where a mesoscale approach describes the actual texture of masonry. Subsequently, a surrogate kriging metamodel is constructed to replace the accurate but computationally expensive numerical descriptions, and global sensitivity analysis is performed to identify the mechanical parameters affecting the most the stiffness and load capacity. Uncertainty propagation is then performed on the surrogate models to estimate the probabilistic distribution of the response parameters of interest. The results provide useful information for risk assessment and management purposes, and shed light on the parameters that control the bridge behaviour and require an accurate characterisation in terms of uncertainty.