Suppose G and H are bipartite graphs and $L: V(G)\to 2^{V(H)}$ induces a partition of $V(H)$ such that the subgraph of H induced between $L(v)$ and $L(v')$ is a matching, whenever $vv'\in E(G)$ . We show for each $\varepsilon>0$ that if H has maximum degree D and $|L(v)| \ge (1+\varepsilon )D/\log D$ for all $v\in V(G)$ , then H admits an independent transversal with respect to L, provided D is sufficiently large. This bound on the part sizes is asymptotically sharp up to a factor $2$ . We also show some asymmetric variants of this result.