Sampling from a target probability distribution is fundamental to modern computational science and machine learning. Sampling is the essence of Monte Carlo integration, enables uncertainty quantification in Bayesian inference, and underlies generative models that have the ability to synthesize convincing text, images, and far beyond. A powerful, emerging approach to sampling is dynamic measure transport (DMT): the idea is to design an ordinary or stochastic differential equation that evolves samples from a tractable reference distribution (e.g., a Gaussian) to the desired target distribution. DMT is state-of-the-art in generative modeling and underlies techniques such as diffusion models and flow-matching, but DMT pipelines for density-driven sampling tasks, as arising in computational chemistry and Bayesian inference, are significantly less developed. In this talk, I will discuss my work to make density-driven DMT a reality via: (1) development of new, gradient-free particle systems for Bayesian sampling, (2) principled design of DMT via PDE-constrained optimization, and (3) scalability through the exploitation of sparse conditional dependence structure.