We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this article do not yield (...) new classes of theories. (shrink)