The laminar breakdown induced by purely travelling crossflow vortices in a three-dimensional flat-plate boundary-layer flow is investigated in detail by means of spatial direct numerical simulations. The base flow considered is generic for an infinite swept wing, with decreasing favourable chordwise pressure gradient and a sweep angle of $45^\circ$. First, the primary downstream growth and nonlinear saturation state of a single crossflow wave are simulated. Secondly, background disturbance pulses are added, and the subsequent mechanisms triggering transition to turbulence in this scenario are identified and analysed in detail. The saturated travelling crossflow vortex is found to give rise to a co-travelling secondary instability not unlike the instability in the much investigated steady crossflow-vortex case, but with characteristic differences. An analysis method with a spanwise Galilean transformation to travel with the primary wave and a consequently adapted timewise/spanwise Fourier decomposition of the disturbance flow enables unambiguous isolation of the various secondary disturbance modes. The resulting flow structures and their dynamics in physical space are visualized.