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The object of this paper is to study estimates of$\epsilon^{-q}W_p(\mu+\epsilon\nu, \mu)$ for small $\epsilon>0$. Here $W_p$ isthe Wasserstein metric on positive measures, $p>1$, $\mu$ is a probabilitymeasure and $\nu$ a signed, neutral measure ($\int d\nu=0$). In [W1] we proveduniform (in $\epsilon$) estimates for $q=1$ provided $\int \phi d\nu$ can becontrolled in terms of the $\int|\nabla\phi|^{p/(p-1)}d\mu$, for any smoothfunction $\phi$.  In this paper we extend the results to the case where such a control fails.This is the case where if, e.g. $\mu$ has a disconnected support, or if thedimension of $\mu$ , $d$ (to be defined) is larger or equal $p/(p-1)$.  In the later case we get such an estimate provided $1/p+1/d\not=1$ for$q=\min(1, 1/p+1/d)$. If $1/p+1/d=1$ we get a log-Lipschitz estimate.  As an application we obtain H\"{o}lder estimates in $W_p$ for curves ofprobability measures which are absolutely continuous in the total variationnorm .  In case the support of $\mu$ is disconnected (corresponding to $d=\infty$) weobtain sharp estimates for $q=1/p$ ("optimal teleportation"): $$\lim_{\epsilon\rightarrow 0}\epsilon^{-1/p}W_p(\mu, \mu+\epsilon\nu) =\|\nu\|_{\mu}$$ where $\|\nu\|_{\mu}$ is expressed in terms of optimaltransport on a metric graph, determined only by the relative distances betweenthe connected components of the support of $\mu$, and the weights of themeasure $\nu$ in each connected component of this support.

From optimal transportation to optimal teleportation