The question I want to ask is close to but not exactly what stated in the title:

Fix a language $L$, it is known that a statement $\sigma$ is universal in the language if whenever $M$ satisfies $\sigma $ and $N$ is a substructure of $M$ then $N$ also satisfy $\sigma$. It is also known that a statement $\sigma$ is existential if whenever $M$ satisfies $\sigma $ and $N$ is an extension of $M$ then $N$ also satisfy $\sigma$.

I can not find generalization of these criteria for formulas with more quantifiers. I wonder why this is the case?