We report upon experimental and analytical investigations of filling box flows in non-uniform porous media characterized by a sudden change in permeability. The porous medium consists of two layers separated by a horizontal permeability jump and is initially filled with light ambient fluid. A line source located at the top of the upper layer supplies dense contaminated fluid that falls toward the bottom of the domain. Two configurations are studied, i.e., a low-permeability layer on top of a high-permeability layer and vice versa. In the former scenario, the flow dynamics are qualitatively similar to the case of a uniform porous medium. Thus, the analytical formulation of Sahu and Flynn (J Fluid Mech 782:455-478, 2015) can be adopted to compute the parameters of interest, e.g., the plume volume flux. In the latter scenario, the flow dynamics are significantly different from those of the uniform porous medium case; after reaching the permeability jump, some fraction of the dense plume propagates horizontally as a pair of oppositely directed interfacial gravity currents. Meanwhile, the remaining fraction of the plume flows downward into the lower layer where it accumulates along the bottom boundary in the form of a deepening layer of discharged plume fluid. Depending on the permeability ratio of the upper and lower layers and the source conditions, the gravity currents may become temporarily arrested after traveling some finite horizontal length. An analytical prediction for this so-called run-out length is derived, motivated, in part, by the immiscible analysis of Goda and Sato (J Fluid Mech 673:60-79, 2011). Finally, a prediction of the filling box time, consisting of the time required to fill the control volume up to the point of contaminated fluid overflow, is made. These predictions are compared with analog experimental measurements. Generally positive agreement is found when the higher-permeability layer is located below the lower-permeability layer. In the opposite circumstance, the agreement is conditional. If the run-out length of the gravity current is less than the horizontal dimensions of the control volume (or tank in case of the experiments), the agreement is good. By contrast, when the run-out length is large, comparatively poor agreement may be realized: In spite of the higher density of the contaminated fluid, it may occupy the entirety of the upper layer before filling the lower layer.