<p>The structural performance of masonry cross vaults on spreading supports is highly nonlinear and discontinuous. Moreover, they are more sensitive to horizontal than vertical support displacement. A GPGPU-parallelised combined finite-discrete element method (FDEM) was employed to evaluate the support displacement that masonry cross vaults could sustain. Dry-joint assumption was held, and each masonry block was discretised into four-node constant strain tetrahedral elements (TET4) connected by six-node cohesive elements (CE6). Fracturing of masonry blocks was not considered, since the failure of cross vaults subjected to horizontal support displacement is largely due to geometrical instability. Numerical examples were presented and validated with experiments, showing that the FDEM was reliable and robust in simulating the behaviour of masonry cross vaults on horizontally moving supports. Support displacement at the time of the first block fall and collapse was recorded. Further parametric discussions on the influence of block weaving pattern, embrace angle of intersecting barrel vaults and rib thickness were performed, and their relevance to the resistance of masonry cross vaults against support displacement was revealed.</p>