  We consider random walks $X,Y$ on a finite graph $G$ with respective
lazinesses $\alpha, \beta \in [0,1]$. Let $\mu_k$ and $\nu_k$ be the $k$-step
transition probability measures of $X$ and $Y$. In this paper, we study the
Wasserstein distance between $\mu_k$ and $\nu_k$ for general $k$. We consider
the sequence formed by the Wasserstein distance at odd values of $k$ and the
sequence formed by the Wasserstein distance at even values of $k$. We first
establish that these sequences always converge, and then we characterize the
possible values for the sequences to converge to. We further show that each of
these sequences is either eventually constant or converges at an exponential
rate. By analyzing the cases of different convergence values separately, we are
able to partially characterize when the Wasserstein distance is constant for
sufficiently large $k$.
