We study the convergence to equilibrium in high dimensions, focusing on explicit bounds on mixing times and the emergence of the cutoff phenomenon for Dyson-Laguerre processes. These are interacting particle systems with non-constant diffusion coefficients, arising naturally in the context of sample covariance matrices. The infinitesimal generator of the process admits generalized Laguerre orthogonal polynomials as eigenfunctions. Our analysis relies on several distances and divergences, including an intrinsic Wasserstein distance adapted to the non-Euclidean geometry of the process. Within this framework, we employ tools from Riemannian geometry and functional inequalities. In particular, we establish exponential decay and derive a regularization inequality for the intrinsic Wasserstein distance via comparison with relative entropy.