Let $G$ be a real Lie group, $Λ\subseteq G$ a lattice, and $X=G/Λ$. We fix a probability measure $μ$ on $G$ and consider the left random walk induced on $X$. It is assumed that $μ$ is aperiodic, has a finite first moment, spans a semisimple algebraic group without compact factors, and has two non mutually singular convolution powers. We show that for every starting point $x\in X$, the $n$-th step distribution $μ^n*δ_{x}$ of the walk weak-$\ast$ converges toward some homogeneous probability measure on $X$.