  We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$,
where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of
finitary permutations of $H$. We show that for any step distribution $\mu$ with
a finite first moment that induces a transient random walk on $H$, the
permutation coordinate of the random walk almost surely stabilizes pointwise.
Our main result states that for $H=\mathbb{Z}$, the above convergence
completely describes the Poisson boundary of the random walk
$(\mathrm{FSym}(\mathbb{Z})\rtimes \mathbb{Z},\mu)$.
