  We combine the method of exchangeable pairs with Stein's method for
functional approximation. As a result, we give a general linearity condition
under which an abstract Gaussian approximation theorem for stochastic processes
holds. We apply this approach to estimate the distance of a sum of random
variables, chosen from an array according to a random permutation, from a
Gaussian mixture process. This result lets us prove a functional combinatorial
central limit theorem. We also consider a graph-valued process and bound the
speed of convergence of the distribution of its rescaled edge counts to a
continuous Gaussian process.
