The observed nonlinear saturation of crossflow vortices in the DLR swept-plate transition experiment, followed by the onset of high-frequency signals, motivated us to compute nonlinear equilibrium solutions for this flow and investigate their instability to high-frequency disturbances. The equilibrium solutions are independent of receptivity, i.e. the way crossflow vortices are generated, and thus provide a unique characterization of the nonlinear flow prior to turbulence. Comparisons of these equilibrium solutions with experimental measurements exhibit strong similarities. Additional comparisons with results from the nonlinear parabolized stability equations (PSE) and spatial direct numerical simulations (DNS) reveal that the equilibrium solutions become unstable to steady, spatial oscillations with very long wavelengths following a bifurcation close to the leading edge. Such spatially oscillating solutions have been observed also in critical layer theory computations. The nature of the spatial behaviour is herein clarified and shown to be analogous to that encountered in temporal direct numerical simulations. We then employ Floquet theory to systematically study the dependence of the secondary, high-frequency instabilities on the saturation amplitude of the equilibrium solutions. With increasing amplitude, the most amplified instability mode can be clearly traced to spanwise inflectional shear layers that occur in the wake-like portions of the equilibrium solutions (Malik et al. 1994 call it ‘mode I’ instability). Both the frequency range and the eigenfunctions resemble recent experimental measurements of Kawakami et al. (1999). However, the lack of an explosive growth leads us to believe that additional self-sustaining processes are active at transition, including the possibility of an absolute instability of the high-frequency disturbances.