We present a general model of the solar wind interaction with a dipolar lunar crustal magnetic anomaly (LMA) using three-dimensional full-kinetic and electromagnetic simulations. We confirm that LMAs may indeed be strong enough to stand off the solar wind from directly impacting the lunar surface, forming a so-called ‘mini-magnetosphere’, as suggested by spacecraft observations and theory. We show that the LMA configuration is driven by electron motion because its scale size is small with respect to the gyro-radius of the solar wind ions. We identify a population of back-streaming ions, the deflection of magnetized electrons via the E × B drift motion, and the subsequent formation of a halo region of elevated density around the dipole source. Finally, it is shown that the presence and efficiency of the processes are heavily impacted by the upstream plasma conditions and, on their turn, influence the overall structure and evolution ofthe LMA system. Understanding the detailed physics of the solar wind interaction with LMAs, including magnetic shielding, particle dynamics and surface charging is vital to evaluate its implications for lunar exploration.