We present a microscopic simulation of convoy electron emission in glancing-angle ion-surface scattering which accounts for both dynamic image interactions and multiple scattering. We find that the position and shape of the convoy peak are determined by rainbow scattering of electrons in the dynamically screened field of the ion. The nonclassical contribution to rainbow scattering is estimated using a uniform semiclassical approach. Core polarization is shown to contribute significantly to dynamical screening. Good agreement with experiment is found for Li+-SnTe (001) co11isions. PACS numbers: 79.20.Nc, 34.50.Fa, 79.20.Rf Electron emission in glancing-angle collisions has become a focus of ion-surface interactions. Electron spectra can provide detailed information about the neutralization dynamics of multiply charged ions, the electronic structure of the surface (surface density of states), and the long-ranged image interactions near the surface. From a fundamental point of view, the study of fast electrons emitted in ion-surface interactions provides an important link between atomic physics and condensed matter physics.Two decades ago a cusp-shaped peak was discovered in the spectrum of electrons arising from ion-atom [1] and ion-solid (transmission) collisions [2]. This cusp is