Bulk metallic glasses (MGs) with tunable plasticity and strength have been reported recently. Using Cu–Zr and Cu–Zr–Al MG models, here we illustrate how and why alloy composition and cooling history influence the initial flow behavior in the early stage of plasticity. Starting from Cu₄₆Zr₅₄, either increasing the Cu concentration, or substituting Al for a few percent of Zr, increases the resistance to the initiation of plastic flow, the softening after the local yielding, and the propensity for strain localization. These effects are shown to be intrinsic to the uniform, fully amorphous MGs and rooted in their internal structure. Our quantitative monitoring of the local environment, especially the role of full-icosahedral clusters in shear transformations, identifies the fertile and resistant structural entities controlling deformation. The structural mechanisms have implications for macroscopic plasticity, and the alloy dependence of the MG structure reveals a microscopic origin underlying the varying mechanical properties.