We study the effects of gravity modulation on the mixing characteristics of two interdiffusing miscible fluids initially in two vertical regions separated by a thin diffusion layer. We formulate the case of general gravity modulation of arbitrary orientation, amplitude $g$ and characteristic frequency $\omega$. For harmonic vertical modulation in two dimensions, the time-dependent Boussinesq equations are solved numerically and the evolution of the interface between the fluids is observed. The problem is governed by six parameters: the Grashof number, $\hbox{\it Gr}\,{=}\,({\Delta\rho}/{\bar{\rho}})g({l^{3}_{\nu}}/{\nu^{2}})$, based on the viscous length scale, $l_{\nu}\,{=}\,\sqrt{{\nu}/{\omega}}$; the Schmidt number, $\hbox{\it Sc}\,{=}\,{\nu}/{D}$; the aspect ratio, $A$; the non-dimensional length of the domain, $l$; the steepness of the initial concentration profile, $\delta$; and the phase angle of the harmonic modulation, $\phi$. When $\phi\,{=}\,0,\;\pi$, we observe four different flow regimes with increasing $\hbox{\it Gr}$: neutral oscillations at the forcing frequency; successive folds which propagate diffusively; localized shear instabilities; and both shear and convective instabilities leading to rapid mixing. In the last regime, the flow is disordered but not chaotic. By varying $\hbox{\it Sc}$, it was determined that the mechanism for the formation of these shear and convective instabilities is inertial. When $\phi \neq 0$ or $\pi$, the flow is similar to a modulated lock exchange flow.