For a pair of i.i.d. sequences of random variables \(X_1,X_2,\ldots \) and \(\tilde{X}_1,\tilde{X}_2,\ldots \) , under suitable conditions we show the existence of an asymptotic expansion in powers of \(m^{-1}\) of the Vaserstein distance \(\mathbb {W}_r(Y_m,\tilde{Y}_m)\) where \(r\ge 1\) , \(Y_m=m^{-1/2}(X_1+\cdots +X_m)\) and \(\tilde{Y}_m\) is defined similarly. This is obtained by applying methods from optimal transport theory to a Cornish–Fisher expansion. The terms in the expansion are determined by the moments of the random variables, and we give some sample calculations of such terms. This is an extension of a central limit problem, to which it reduces when the means are 0 and \(\tilde{Y}_1\) is normal with the same variance as \(X_1\) . We also describe an application to the question of eventual monotonicity in m of the sequence \(\mathbb {W}_r(Y_m,\tilde{Y}_m)\) , related to a question of Villani for the case \(r=2\) .