We show that in an N-partite system, where each subsystem is subject to a local completely positive map, total correlations cannot, at later times, exceed their initial value. We then analyze the dynamics of correlations in a pair of noninteracting qubits under the local action of random classical external fields. We find decreasing, increasing and constant plateaux of both quantum and classical correlations and also disappearance and revival of entanglement, similarly to what occurs in independent quantum environments. These revivals cannot here be explained by exchange of correlations with the environment. We instead associate all the above phenomena with the intrinsic non-Markovianity of the map, drawing the counter-intuitive conclusion that they may occur irrespective of whether the external noise is quantum or classical.