Tiling with monochromatic bipartite graphs of bounded maximum degree

We prove that for any $$, there exists a constant $$ such that the following is true. Let $$ be an infinite sequence of bipartite graphs such that $$ and $$ hold for all $$. Then, in any $$-edge-coloured complete graph $$, there is a collection of at most $$ monochromatic subgraphs, each of which is isomorphic to an element of $$, whose vertex sets partition $$. This proves a conjecture of Corsten and Mendonça in a strong form and generalises results on the multi-colour Ramsey numbers of bounded-degree bipartite graphs. It also settles the bipartite case of a general conjecture of Grinshpun and Sárközy.