In this paper, we address the problem of robust stability and set invariance of switched linear parameter varying (LPV) systems affected by bounded disturbances. A switched LPV system is a collection of single LPV subsystems with a switching rule that decides which subsystem is active at each time. This framework is useful for LPV systems with large parameter variation range for which it may be difficult to assess or guarantee stability over the whole parameter range. Our first contribution is to derive a result for a single LPV system that extends and generalises previous work for switched systems. Our second, and main, contribution is to apply this seed result in the derivation of dwell-time-type conditions on the switched-LPV switching rule, which ensure robust closed-loop stability and set invariance of the trajectories of the switched-LPV system across the whole parameter range. As an application of the results, we consider the problem of reference tracking for LPV systems using switched LPV state feedback and feedforward from an LPV reference system. Two examples are presented to illustrate the results: a two-mass-spring system with a varying spring characteristic and the nonlinear model of a coupled-tank system which is embedded into a switched LPV system description.