AbstractWe introduce a holomorphic sheafℰ${\mathcal{E}}$on a Sasakian manifoldSand study two new notions of stability forℰ${\mathcal{E}}$along the Sasaki–Ricci flow related to the ‘jumping up’ of the number of global holomorphic sections ofℰ${\mathcal{E}}$at infinity. First, we show that if the Mabuchi K-energy is bounded below, the transverse Riemann tensor is bounded inC0${C^{0}}$along the flow, and theC∞${C^{\infty}}$closure of the Sasaki structure onSunder the diffeomorphism group does not contain a Sasaki structure with strictly more global holomorphic sections ofℰ${\mathcal{E}}$, then the Sasaki–Ricci flow converges exponentially fast to a Sasaki–Einstein metric. Secondly, we show that if the Futaki invariant vanishes, and the lowest positive eigenvalue of the∂¯${\bar{\partial}}$Laplacian on global sections ofℰ${\mathcal{E}}$is bounded away from zero uniformly along the flow, then the Sasaki–Ricci flow converges exponentially fast to a Sasaki–Einstein metric.