Let G be a real Lie group, Λ ≤ 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 ∈ 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.