  The Wasserstein distance $\mathcal{W}_p$ is an important instance of an
optimal transport cost. Its numerous mathematical properties as well as
applications to various fields such as mathematical finance and statistics have
been well studied in recent years. The adapted Wasserstein distance
$\mathcal{A}\mathcal{W}_p$ extends this theory to laws of discrete time
stochastic processes in their natural filtrations, making it particularly well
suited for analyzing time-dependent stochastic optimization problems.
  While the topological differences between $\mathcal{A}\mathcal{W}_p$ and
$\mathcal{W}_p$ are well understood, their differences as metrics remain
largely unexplored beyond the trivial bound $\mathcal{W}_p\lesssim
\mathcal{A}\mathcal{W}_p$. This paper closes this gap by providing upper bounds
of $\mathcal{A}\mathcal{W}_p$ in terms of $\mathcal{W}_p$ through investigation
of the smooth adapted Wasserstein distance. Our upper bounds are explicit and
are given by a sum of $\mathcal{W}_p$, Eder's modulus of continuity and a term
characterizing the tail behavior of measures. As a consequence, upper bounds on
$\mathcal{W}_p$ automatically hold for $\mathcal{AW}_p$ under mild regularity
assumptions on the measures considered. A particular instance of our findings
is the inequality $\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$ on the
set of measures that have Lipschitz kernels.
  Our work also reveals how smoothing of measures affects the adapted weak
topology. In fact, we find that the topology induced by the smooth adapted
Wasserstein distance exhibits a non-trivial interpolation property, which we
characterize explicitly: it lies in between the adapted weak topology and the
weak topology, and the inclusion is governed by the decay of the smoothing
parameter.
