During the last decade several stochastic stratified processes were introduced in the probability literature. They include the fiber Brownian motion introduced by Bass and Burdzy as a process that switches between two-dimensional and one-dimensional evolution, a Markov process on a `whiskered sphere' described by Sowers, processes on trees like the Walsh process, the Evans process, spider martingales, and Brownian snakes. We study the rate of weak convergence to one such process. Using a Wasserstein-type metric as the distance that metrizes weak convergence, we show that the rate of this convergence can be controlled by the rates of convergence of other related processes. For a related classical example of a Hamiltonian system on a cylinder, the corresponding estimates on the rate of convergence are obtained using two different methods. One of these methods reveals an explicit expression for the Wasserstein distance in the classical case.