Quantifying the ways in which local particle rearrangements contribute to macroscopic plasticity is one of the fundamental pursuits of granular mechanics and soft matter physics. Here we examine local rearrangements that occur naturally during the deformation of three samples of 3D granular materials subjected to distinct boundary conditions by employing in situ x-ray measurements of particle-resolved structure and stress. We focus on five distinct rearrangement measures, their statistics, interrelationships, contributions to macroscopic deformation, repeatability, and dependence on local structure and stress. Our most significant findings are that local rearrangements (1) are correlated on a scale of three to four particle diameters, (2) exhibit volumetric strain-shear strain and nonaffine displacement-rotation coupling, (3) exhibit correlations that suggest either rearrangement repeatability or that rearrangements span multiple steps of incremental sample strain, and (4) show little dependence on local stress but correlate with quantities describing local structure, such as porosity. Our results are presented in the context of relevant plasticity theories and are consistent with recent findings suggesting that local structure may play at least as important of a role as local stress in determining the nature of local rearrangements.