  We prove that independent families of permutation invariant random matrices
are asymptotically free over the diagonal, both in probability and in
expectation, under a uniform boundedness assumption on the operator norm. We
can relax the operator norm assumption to an estimate on sums associated to
graphs of matrices, further extending the range of applications (for example,
to Wigner matrices with exploding moments and so the sparse regime of the
Erd\H{o}s-R\'{e}nyi model). The result still holds even if the matrices are
multiplied entrywise by bounded random variables (for example, as in the case
of matrices with a variance profile and percolation models).
