Are there any results on the following "generalized matching" problems?
In http://www.sciencedirect.com/science/article/pii/S0012365X13003543 , we use exactly this process of picking the largest possible clique, then the next largest available clique, and so on (without repeating any vertices). When applied to a cograph $G$, this algorithm forms a cluster graph subgraph of $G$ with the max number of edges possible. The paper contains some combinatorial results on sizes of cliques obtained during such a process.

This paper http://publicaciones.dc.uba.ar/Publications/2015/BDNV15/BDNV15.pdf generalizes our result to a class of graphs beyond cographs.

I'm not sure which way you want to generalize matchings ... towards triangle covers or clique covers, but either way, it seems you will be moving from P-territory into NP-complete land. Unless you restrict your initial graph to certain structural classes.