  We determine when there exists a ribbon rational homology cobordism between
two connected sums of lens spaces, i.e. one without $3$-handles. In particular,
we show that if a lens space $L$ admits a ribbon rational homology cobordism to
a different lens space, then $L$ must be homeomorphic to $L(n,1)$, up to
orientation-reversal. As an application, we classify ribbon $\chi$-concordances
between connected sums of $2$-bridge links. Our work builds on Lisca's work on
embeddings of linear lattices.
