We study porous medium (or, equivalently, Hele-Shaw) flows with a moving boundary which arise as scaling (i.e., continuous) limits of certain discrete aggregation models. Specifically, we study the scaling limit of the internal DLA model with a killing and a one-way passing condition for particles on the negative semi-axis. These models were recently studied by L. Levine and Y. Peres. We give an exact self-similar solution for the corresponding porous medium flow in the first case (which matches perfectly the numerical data obtained by L. Levine and Y. Peres), and derive moment properties of such solution in the second case.