Many instances of slow-moving multiphase flows exist in our world, from the movement of oil and water under the earth’s surface to the flow of blood under our skin. In the case of droplets flowing through porous media, not all will make it through. Utilizing the boundary-integral simulation method, which requires only interfaces to be meshed, we show what happens inside drops that get stuck. Several new computational techniques are highlighted, including multimesh desingularization to allow drops to interact with complex-shaped particles, and 3D contour advection to show how interfaces within drops deform over time. Thanks to Stefan Kartenberg and septahelix for licensing their music under Creative Commons.