The normalised partial sums of values of a nonnegative multiplicative function over divisors with appropriately restricted lengths of a random permutation from the symmetric group define trajectories of a stochastic process. We prove a functional limit theorem in the Skorokhod space when the permutations are drawn uniformly at random. Furthermore, we show that the paths of the limit process almost surely belong to the space of continuous functions on the unit interval and, exploiting results from number-theoretic papers, we obtain rather complex formulas for the limits of joint power moments of the process.