  We show that for any $2$-local colouring of the edges of the balanced
complete bipartite graph $K_{n,n}$, its vertices can be covered with at
most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of
any complete or balanced complete bipartite $r$-locally coloured graph with
$O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local
bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of
the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.
