We estimate the Wasserstein type distance between two continuous distributions F and G on R such that the set {F = G} is a finite union of intervals, possibly empty or R. The positive cost function ρ is not necessarily symmetric and the sample may come from any joint distribution H on R 2 having marginals F and G with light enough tails with respect to ρ. The rates of weak convergence and the limiting distributions are derived in a wide class of situations including the classical distances W1 and W2. The key new assumption in the case F = G involves the behavior of ρ near 0, which we assume to be regularly varying with index ranging from 1 to 2. Rates are then also regularly varying with powers ranging from 1/2 to 1 also affecting the limiting distribution, in addition to H.