Let $G$ be a connected simple real Lie group, $Λ_{0}\subseteq G$ a lattice and $Λ\unlhd Λ_{0}$ a normal subgroup such that $Λ_{0}/Λ\simeq \mathbb{Z}^d$. We study the drift of a random walk on the $\mathbb{Z}^d$-cover $Λ\backslash G$ of the finite volume homogeneous space $Λ_{0}\backslash G$. This walk is defined by a Zariski-dense compactly supported probability measure $μ$ on $G$. We first assume the covering map $Λ\backslash G\rightarrow Λ_{0}\backslash G$ does not unfold any cusp of $Λ_{0}\backslash G$ 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.