  In the infinite regular tree $\mathbb{T}_{q+1}$ with $q \in \mathbb{Z}_{\ge
2}$, we consider families $\{\mu_u^n\}$, indexed by vertices $u$ and
nonnegative integers ("discrete time steps") $n$, of probability measures such
that $\mu_u^n(v) = \mu_{u'}^n(v')$ if the distances $\operatorname{dist}(u,v)$
and $\operatorname{dist}(u',v')$ are equal. Let $d$ be a positive integer, and
let $X$ and $Y$ be two vertices in the tree which are at distance $d$ apart. We
compute a formula for the transportation distance $W_1\!\left( \mu_X^n, \mu_Y^n
\right)$ in terms of generating functions. In the special case where $\mu_u^n =
\mathfrak{m}_u^n$ are measures from simple random walks after $n$ time steps,
we establish the linear asymptotic formula $W_1\!\left( \mathfrak{m}_X^n,
\mathfrak{m}_Y^n \right) = An + B + o(1)$, as $n \to \infty$, and give the
formulas for the coefficients $A$ and $B$ in closed forms. We also obtain
linear asymptotic formulas in the cases of spheres and uniform balls as the
radii tend to infinity. We show that these six coefficients (two from simple
random walks, two from spheres, and two from uniform balls) are related by
inequalities.
