The interplay of reaction and diffusion processes can trigger localized spatiotemporal patterns when two solutions containing separate reactants A and B of an oscillating reaction are put in contact. Using the Brusselator, a classical model for chemical oscillations, we show numerically that localized waves and Turing patterns as well as reaction-diffusion (RD) patterns due to an interaction between these two kinds of modes can develop in time around the reactive contact zone depending on the initial concentration of reactants and diffusion coefficients of the intermediate species locally produced. We further explore the possible hydrodynamic destabilization of an initially buoyantly stable stratification of such an A + B → oscillator system, when the chemical reaction provides a buoyant periodic forcing via localized density changes. Guided by the properties of the underlying RD dynamics, we predict new chemo-hydrodynamic instabilities on the basis of the dynamic density profiles which are here varying with the concentration of one of the intermediate species of the oscillator. Nonlinear simulations of the related reaction-diffusion-convection equations show how the active coupling between the localized oscillatory kinetics and buoyancy-driven convection can induce pulsatile convective fingering and pulsatile plumes as well as rising or sinking Turing spots, depending on the initial concentration of the reactants and their contribution to the density.