  Let $\Gamma$ be a countable discrete group, $H$ a lcsc totally disconnected
group and $\rho : \Gamma \rightarrow H$ a homomorphism with dense image. We
develop a general and explicit technique which provides, for every compact open
subgroup $L < H$ and bi-$L$-invariant probability measure $\theta$ on $H$, a
Furstenberg discretization $\tau$ of $\theta$ such that the Poisson boundary of
$(H,\theta)$ is a $\tau$-boundary. Among other things, this technique allows us
to construct examples of finitely supported random walks on certain lamplighter
groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are
prime, but not $L^p$-irreducible for any $p \geq 1$, answering a conjecture of
Bader-Muchnik in the negative. Furthermore, we give an example of a countable
discrete group $\Gamma$ and two spread-out probability measures $\tau_1$ and
$\tau_2$ on $\Gamma$ such that the boundary entropy spectrum of
$(\Gamma,\tau_1)$ is an interval, while the boundary entropy spectrum of
$(\Gamma,\tau_2)$ is a Cantor set.
