Let $G$ be a real Lie group, $Λ\leq G$ a lattice, and $Ω=G/Λ$. We study the equidistribution properties of the left random walk on $Ω$ induced by a probability measure $μ$ on $G$. It is assumed that $μ$ has a finite first moment, and that the Zariski closure of the group generated by the support of $μ$ in the adjoint representation is semisimple without compact factors. We show that for every starting point $x\in Ω$, the $μ$-walk with origin $x$ has no escape of mass, and equidistributes in Cesàro averages toward some homogeneous measure. This extends several fundamental results due to Benoist-Quint and Eskin-Margulis for walks with finite exponential moment.