Rainbow matchings in bipartite multigraphs

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\lfloor \frac{k+2}{k+1}n\rfloor -(k+1)$$ matchings $M_1,\dots ,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$’s. Several choices of parameters relate to known results and conjectures.