In this paper, the approach of the control bumps limitation is generalised and applied to the problem of H-infinity bumpless transfer for a class of switched linear parameter-varying (LPV) systems. The H-infinity bumpless transfer problem is to design a controller with the amplitude limitation and a parameter and modes-dependent switching law to reduce control bumps of the switched LPV systems and satisfy an H-infinity performance index. The key point is to guarantee the bumps limitation constraint of the switched LPV systems by dual design of controllers and a switching law, even though the constraint for each subsystem is unsatisfied. First, a set of switching signals depending on parameters and modes are designed, and a family of switched LPV controllers with the bumps limitation objective are developed via multiple parameter-dependent Lyapunov functions. Then, a sufficient condition ensuring the H-infinity bumpless transfer performance for a switched LPV system is presented in the format of numerically tractable conditions. Finally, the effectiveness of the proposed control design scheme is illustrated by its application to the control design of an aero-engine.