The tilting mechanism, which generates differential rotation in two-dimensional turbulent convection, is shown to produce relaxation oscillations in the mean flow energy integral and bursts in the global fluctuation level, akin to Lotka-Volterra oscillations. The basic reason for such behavior is the unidirectional and conservative transfer of kinetic energy from the fluctuating motions to the mean component of the flows, and its dissipation at large scales. Results from numerical simulations further demonstrate the intimate relation between these low-frequency modulations and the large-scale intermittency of convective turbulence, as manifested by exponential tails in single-point probability distribution functions. Moreover, the spatio-temporal evolution of convective structures illustrates the mechanism triggering avalanche events in the transport process. The latter involves the overlap of delocalized mixing regions when the barrier to transport, produced by the mean component of the flow, transiently disappears.