  We will prove a decomposition for Wasserstein geodesics in the following
sense: let $(X,d,m)$ be a non-branching metric measure space verifying
$\mathsf{CD}_{loc}(K,N)$ or equivalently $\mathsf{CD}^{*}(K,N)$. We prove that
every geodesic $\mu_{t}$ in the $L^{2}$-Wasserstein space, with $\mu_{t} \ll
m$, is decomposable as the product of two densities, one corresponding to a
geodesic with support of codimension one verifying $\mathsf{CD}^{*}(K,N-1)$,
and the other associated with a precise one dimensional measure, provided the
length map enjoys local Lipschitz regularity. The motivation for our
decomposition is in the use of the component evolving like $\mathsf{CD}^{*}$ in
the globalization problem.
  For a particular class of optimal transportation we prove the linearity in
time of the other component, obtaining therefore the global $\mathsf{CD}(K,N)$
for $\mu_{t}$. The result can be therefore interpret as a globalization theorem
for $\mathsf{CD}(K,N)$ for this class of optimal transportation, or as a
``self-improving property'' for $\mathsf{CD}^{*}(K,N)$. Assuming more
regularity, namely in the setting of infinitesimally strictly convex metric
measure space, the one dimensional density is the product of two differentials
giving more insight on the density decomposition.
