Generalizing a conjecture of Aharoni, Joos and Kim asked the following intriguing question. Let H be a graph on m edges, and let G_i (1<=i<=m) be a sequence of m graphs on the common vertex set [n]. What is the weakest minimum degree restriction we can impose on each G_i to guarantee a rainbow copy of H? Joos and Kim answered this question when H is a Hamilton cycle or a perfect matching. We provide an asymptotic answer when H is an F-factor, or a spanning tree with maximum degree o(n/log n). This is joint work with Richard Montgomery and Yani Pehova.