The answers above already say it all.Still, if you are a visual person, have a look at the following optimal coupling/transport plan (or to be precise, a sample thereof. Note that the colour's strength is a function of the empirical density.) between a univariate $T$ distribution with 3 degrees of freedom and a standard Gaussian. You can really interpret this as "to reshape optimally a $T_3$ distribution to get a Gaussian, I need to push the mass contained in the tails towards the middle". To see this, observe that an infinitesimal volume element around -4 for the student will approximately be transported to -2. This is a particularity of the 1D case, but the intuition "holds" for higher dimensions.
