  Let $G$ be a connected simple real Lie group, $\Lambda_{0}\subseteq G$ a
lattice and $\Lambda \unlhd \Lambda_{0}$ a normal subgroup such that
$\Lambda_{0}/\Lambda\simeq \mathbb{Z}^d$. We study the drift of a random walk
on the $\mathbb{Z}^d$-cover $\Lambda\backslash G$ of the finite volume
homogeneous space $\Lambda_{0}\backslash G$. This walk is defined by a
Zariski-dense compactly supported probability measure $\mu$ on $G$. We first
assume the covering map $\Lambda\backslash G\rightarrow \Lambda_{0}\backslash
G$ does not unfold any cusp of $\Lambda_{0}\backslash G$ and compute the drift
at \emph{every} starting point. Then we remove this assumption and describe the
drift almost everywhere. The case of hyperbolic manifolds of dimension 2 stands
out with non-converging type behaviors. The recurrence of the trajectories is
also characterized in this context.
