We examine the transient buoyancy-driven flow in a ventilated filling box that is subject to a continuous supply of buoyancy. A rectangular box is considered and the buoyancy input is represented as a turbulent plume, or as multiple non-interacting plumes, rising from the floor. Openings in the base and top of the box link the interior environment with a quiescent exterior environment of constant and uniform density. A theoretical model is developed to predict, as functions of time, the density stratification and the volume flow rate through the openings leading to the steady state. Comparisons are made with the results of small-scale analogue laboratory experiments in which saline solutions and fresh water are used to create density differences. Two characteristic timescales are identified: the filling box time ($T_f$), proportional to the time taken for fluid from a plume to fill a closed box; and the draining box time ($T_d$), proportional to the time taken for a ventilated box to drain of buoyant fluid. The timescale for the flow to reach the steady state depends on these two timescales, which are functions of the box height $H$ and cross-sectional area $S$, the ‘effective’ opening area $A^*$, and the strength, number and distribution of the buoyancy inputs. The steady-state flow is shown to be characterized by the ratio of these timescales ($\mu\,{=}\,T_d/T_f$) which is equivalent to the dimensionless vent area $A^*/H^2$. A feature of these flows is that for $\mu\,{>}\,\mu_c$ the depth of the buoyant upper layer may exceed, or ‘overshoot’, the steady layer depth during the initial transient. The value of $\mu_c$ is determined for both line and point-source plumes, and the sensitivity of the developing flow to the distribution of buoyancy input assessed.