  In systems where electrons form both dispersive bands and small local spins,
we show that changes of the spin configuration can tune the bands through a
Lifshitz transition, resulting in a continuous metal-insulator transition
associated with a progressive change of the Fermi surface topology. In contrast
to a Mott-Hubbard and Slater pictures, this spin-driven Lifshitz transition
appears in systems with small electron-electron correlation and large
hybridization. We show that this situation is realized in 5$d$ distorted
perovskites with an half-filled $t_{2g}$ bands such as NaOsO$_3$, where the
strong $p-d$ hybridization reduces the local moment, and spin-orbit coupling
causes a large renormalization of the electronic mobility. This weakens the
role of electronic correlations and drives the system towards an itinerant
magnetic regime which enables spin-fluctuations.
