We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.