For optimizing a non-convex function in finite dimension, a method is to add Brownian noise to a gradient descent, allowing for transitions between basins of attractions of different minimizers. To adapt this for optimization over a space of probability distributions requires a suitable noise. For this purpose, we introduce here a simple stochastic process where a number of moments of the distribution are following a chosen finite-dimensional diffusion process, generalizing some previous studies where the expectation of the measure is subject to a Brownian noise. The process may explode in finite time, for instance when trying to force the variance of a distribution to behave like a Brownian motion. We show, up to the possible explosion time, well-posedness and propagation of chaos for the system of mean-field interacting particles with common noise approximating the process.