Kondo insulators are emerging as a simplified setting to study both magnetic and insulator-to-metal quantum phase transitions. Here, we study the half-filled Anderson lattice model defined on a magnetically frustrated Shastry-Sutherland geometry. We determine a “global” phase diagram that applies to both the local-moment and intermediate-valence regimes. This provides the theoretical basis for understanding how tuning a Kondo insulator by external parameters can close its hybridization gap, liberate the local-moment spins from the conduction electrons, and lead to a magnetically correlated metal. We also calculate the momentum distribution of the single-particle excitations in the Kondo insulating state, and show how Fermi-surface-like features emerge as a precursor to the actual Fermi surfaces of the Kondo-destroyed metals. The implications for an incipient Fermi surface and quantum phase transitions of Kondo insulators including SmB6 are discussed.