We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with drift, and a congested crowd motion model originally proposed by Maury et al (2010 Math. Models Methods Appl. Sci. 20 1787-821). We first use viscosity solutions to show that the porous medium equation solutions converge to the Hele-Shaw solution as m → ∞ provided the drift potential is strictly subharmonic. Next, using the gradient-flow structure of both the porous medium equation and the crowd motion model, we prove that the porous medium equation solutions also converge to the congested crowd motion as m → ∞. Combining these results lets us deduce that in the case where the initial data to the crowd motion model is given by a patch, or characteristic function, the solution evolves as a patch that is the unique solution to the Hele-Shaw problem. While proving our main results we also obtain a comparison principle for solutions with the minimizing movement scheme based on the Wasserstein metric, of independent interest.