Results of laboratory experiments are presented in which a fixed volume of homogeneous fluid is suddenly released into another fluid of slightly lower density, over a horizontal thin metallic grid placed a given distance above the solid bottom of a rectangular-cross-section channel. Dense liquid develops as a gravity current over the grid at the same time as it partially flows downwards. The results show that the gravity current loses mass at an exponential rate through the porous substrate with a time constant τ; the front velocity and the head of the current also decrease exponentially. The loss of mass dominates the flow and, in contrast to gravity currents running over solid bottoms, no self-similar inertial regime seems to be developed. A simple model is introduced to explain the scaling law of the loss of mass and the evolution of the front position. The flow evolution depends on the characteristic time of the initial (slumping) phase and the time constant τ, related to the initial conditions and the permeability of the porous substrate, respectively. Qualitative comparisons with other gravity currents with loss of mass, such as particle-driven gravity currents, are provided.