Suppose that $\Phi(x,y)$ is a formula in the language of set theory so that for each natural number $n$, the axiom $\exists x\Phi(n,x)$ is a large cardinal axiom (for example consider $n$-huge cardinals). Then does $\text{Con}(ZFC+\exists x\Phi(n,x))$ for all $n$ imply $\text{Con}(ZFC+\forall n\exists x\Phi(n,x))$ (i.e. informally is the large cardinal heirarchy continuous in this sense or are there jumps in the large cardinal heirarchy)? Could there be a large cardinal axiom whose consistency strength is greater than each axiom $\exists x\Phi(n,x)$ for all natural numbers $n$ but whose consistency strength is strictly less than $\forall n\exists x\Phi(n,x)$?

Is it possible that there is a large cardinal axiom $\Psi$ so that $ZFC\models\text{Con}(\Psi)\leftrightarrow\forall n\in\omega\,\text{Con}(\Phi(n,x))$? What would the answer to these questions be if we replaced $\text{Con}(\Phi)$ with the existence of a transitive model that satisfies $\Phi$ or some other strengthening of mere consistency?

If one needs a formalization of what is meant by a large cardinal axiom, one can use this formalization due to Woodin mentioned in Mohammad Golshani's answer here or something similar to that.

I am also be interested in discontinuities in the large cardinal heirarchy by sequences of cofinality other than $\omega$ (for instance, for a sequence of cofinality $\aleph_{1},\mathfrak{c}$ or the first inaccessible cardinal), but I do not know of a nice way to formalize this notion.