Recently, the authors explored the effectiveness of using an inerter-based device to mitigate pounding induced damage between two adjacent bridge structures, and the advantages of this system were illustrated by comparing its performances with the conventional viscous dampers. However, the seismic excitations at the supports of the adjacent bridge structures were assumed to be the same in that study, i.e. only uniform ground motion was considered. In reality, ground motions at different locations of a structure are inevitably different, and spatially varying ground motions (SVGMs) can influence the structural seismic responses significantly. Extensive research works have been carried out to investigate the influence of SVGMs on the seismic responses of engineering structures. However, no open literature reported the results when an inerter-based device is involved. In this study, the motion equations of two adjacent bridges coupled by a tuned inerter damper (TID) are firstly derived, and the seismic responses of the adjacent bridges with and without control are calculated. In particular, the parameters that significantly influence SVGMs, i.e. wave passage effect and coherency loss effect, are discussed. •Equations for adjacent bridges coupled by a TID subjected to SVGMs are derived.•The influences of the key parameters of SVGMs are systematically discussed.•TID decreases the sensitivity of relative displacement to the frequency ratio of adjacent structures.