We report x-ray photon correlation spectroscopy (XPCS) experiments on a series of concentrated oil-in-water nanoemulsions with varying droplet volume fraction subjected to in situ steady-state large-amplitude oscillatory shear (LAOS). The shear strain causes periodic echoes in the x-ray speckle patterns that lead to peaks in the intensity autocorrelation function. Above an onset strain amplitude that depends on nanoemulsion concentration, the peaks become attenuated, signaling spatially heterogeneous, shear-induced droplet dynamics. These dynamics include irreversible rearrangements among the droplets that occur in some regions of the nanoemulsions during a given shear cycle and residual strain-like displacements in those regions that do not rearrange. The wave-vector dependence of the peak attenuation indicates a power-law distribution in the size of regions undergoing shear-induced rearrangement that is similar to that observed previously in LAOS-XPCS measurements on concentrated nanocolloidal gels. The values of the onset strains for rearrangement correlate with the concentration-dependent macroscopic yielding behavior of the nanoemulsions. Specifically, they occur below the strains at which the nanoemulsions become effectively fluidized and, except for the lowest-concentration nanoemulsion in the study, significantly above the threshold strain for nonlinear rheological response.