Anti-Ramsey theory was initiated in 1975 by Erdős, Simonovits and Sós, inspiring hundreds of publications since then. The present work is the third and last piece of our trilogy in which we introduce a far-reaching generalization via the following two functions for any graph $G$ and family ${\cal F}$ of graphs: If $K_2 \in {\cal F}$, let $f(n,G|{\cal F})$ be the smallest integer $k$ such that every edge coloring of $K_n$ with at least $k$ colors forces a copy of $G$ in which all color classes are members of ${\cal F}$. If $K_2 \notin {\cal F}$, let $g(n,G|{\cal F})$ be the largest integer $k$ for which there exists an edge coloring of $K_n$ using exactly $k$ colors, such that every copy of $G$ contains an induced color class which is a member of ${\cal F}$. We develop methods suitable for deriving asymptotically tight results for the $f$-function and the $g$-function for many combinations of $G$ and ${\cal F}$. The preceding parts of the trilogy are arXiv: 2405.19812 and 2408.04257, published in Discrete Applied Math. Vol. 363 and Mathematics Vol. 12:23, respectively.