We develop a novel interacting-particle method for sampling from non-Gaussian distributions. As a first step, we propose a new way to derive the consensus-based sampling (CBS) algorithm, starting from ensemble-preconditioned Langevin diffusions. We approximate the target potential by its Moreau envelope, such that the gradient in the Langevin equation can be replaced by a proximal operator. We then approximate the proximal operator by a weighted mean, and finally assume that the initial and target distributions are Gaussian, resulting in the CBS dynamics. If we keep only those approximations that can be justified in the non-Gaussian setting, the result is a new interacting-particle method for sampling, which we call localized consensus-based sampling. We prove that our algorithm is affine-invariant and exact for Gaussian distributions in the mean-field setting. Numerical tests illustrate that localized CBS compares favorably to alternative methods in terms of affine-invariance and performance on non-Gaussian distributions.