This talk concerns recent work on a class of PDEs with linear and nonlinear diffusion, including the heat equation, fast diffusion equations, and height constrained transport. We develop and prove convergence of a nonlocal approximation for such equations. This gives rise to a deterministic particle numerical method for these PDEs, as well as a novel particle method for sampling a wide range of probability measures. In this talk, I will highlight the how our convergence arguments take advantage of both the Wasserstein and the dual Sobolev gradient flow structures of the PDEs under consideration. Based on joint work with Katy Craig and Matt Jacobs.