For a lower bound on the stability, consider the example A = [0,1]^n and B = [0,1+delta]^(n/2) times [0,1/(1+delta)]^(n/2). By choosing delta appropriately, the stability in terms of the measure of the symmetric difference scales like n^2 (although the stability in terms of the Wasserstein distance remains fixed as n grows). This scaling is actually the worst that is known, so it is possible that the bound of Figalli, Maggi and Pratelli (which grows like n^7) could be improved. A joint work with Bo'az Klartag