Motivated by recent observations of quantum oscillations of magnetization in a Kondo insulator (SmB_6), we investigate the Kondo lattice model of spin-1/2 local moments coupled to the conduction electrons at half-filling for its orbital response to magnetic field. The key finding of our study is that the quantum oscillations of magnetization in Kondo insulators occur as a bulk phenomenon, which manifests itself through the inversion of a Hofstadter - quantized dispersion of the gaped charge quasi-particles whose chemical-potential surface these oscillations measure . We find this through a minimal effective dynamics, in a certain canonical representation of electrons , that appropriately describes the Kondo insulating state, and reveals important new features such as the inversion and Lifshitz-like transition for the charge quasi-particles. We also predicts these oscillations to occur in the spin-density wave insulators .