•Propose a framework for simulating maglev trains running over transitional viaducts.•The transitional viaduct is assumed to be realistically deformable, instead of rigid.•Both curve radius and height difference of the inner and outer rails are considered.•The motion of train running over a transitional track is expressed by trajectory coordinates.•Analysis results are verified by the experimental ones for the Shanghai maglev line.•Vehicle rolling motions affect significantly the ride quality on transitional tracks. This paper presents a general framework for the dynamic analysis of high-speed maglev trains running on elastic transitional viaducts, a complicated problem that lacks an efficient analysis tool. Unlike the straight and circular tracks, both the curve radius (CR) and height difference (HD) between the outer and inner rails vary along the transitional track. To start, the motion of each vehicle of the train running over a transitional track is expressed by a set of trajectory coordinates, with their orientations given in the Euler angles as functions of the CR and HD. The elastic transitional viaduct is simulated by finite elements in the global coordinates. For the maglev trains, the electromagnet force-air gap model is used to account for the interaction and coupling effect between the moving vehicles and supporting viaduct. By applying the proposed framework to the Shanghai maglev line (SML), the dynamic responses of the maglev vehicles running on the transitional viaduct are studied and compared with those on the straight and circular viaducts. Moreover, the effects of transitional track length and cant deficiency on the coupled system are investigated. The results show that for vehicles running on the transitional track, the levitation forces and vehicle’s angular velocity are highly related to the rate of change of HD of the track. The length of the transitional viaduct affects significantly all the vehicle’s responses. The increase of deficient cant angle causes a sharp increase in the responses of the transitional viaduct in both the radial and rotational directions.