Masonry structures are often characterized by complex, non-planar geometries. This is also the case for historical and monumental structures. Here we investigate the dynamic behaviour of non-standard, curvilinear masonry geometries, such as vaults, subjected to blast loading. We use the Discrete Element Method (DEM) for modelling the dynamic structural response to explosions. The approach allows considering the detailed mechanical and geometrical characteristics of masonry, as well as the inherent coupling between the in-and out-of-plane motion. The proposed modelling approach is validated with existing experimental tests in the case of planar masonry geometries, walls, subjected to far-field explosions. The DEM model is found to satisfactorily capture the dynamic response of the system and the form of failure within the body of the masonry structure. Then the response of an emblematic curved masonry structure subjected to blast loading is investigated. The influence of various micro-mechanical parameters, such as the dilatancy angle, the tensile strength and the cohesion of the masonry joints on the overall dynamic structural response of the system is explored. The effect of the size of the building blocks is also studied. Masonry joints with zero dilatancy lead to increased out-of-plane deformations and reduced membrane ones, with respect to associative case. Cohesion and tensile strength are found to have negligible influence on the structural response. The size of the building blocks shapes the overall strength of the system. Finally, the performance of a discrete model with infinitely rigid blocks is explored and evaluated through detailed comparisons of the parametric numerical tests using deformable blocks. The rigid blocks model predictions, for the loading conditions and geometries here investigated, are found to be affected by the rotational locking effect.