A generalization of the famous Caccetta–Häggkvist conjecture, suggested by Aharoni, is that any family (Formula presented.) of sets of edges in (Formula presented.), each of size (Formula presented.), has a rainbow cycle of length at most (Formula presented.). In works by the author with Aharoni and by the author with Aharoni, Berger, Chudnovsky, and Zerbib, it was shown that asymptotically this can be improved to (Formula presented.) if all sets are matchings of size 2, or all are triangles. We show that the same is true in the mixed case, that is, if each (Formula presented.) is either a matching of size 2 or a triangle. We also study the case that each (Formula presented.) is a matching of size 2 or a single edge, or each (Formula presented.) is a triangle or a single edge, and in each of these cases we determine the threshold proportion between the types, beyond which the rainbow girth goes from linear to logarithmic.