I want to know if there are fairly simple combinatorial necessary conditions for when a direct limit of ultrapowers of $V$ is well-founded similar to $\sigma$-completeness. By combinatorial, I mean that these conditions are conditions on the ultrafilters instead of the elementary embeddings they produce.
For simplicity, we shall formulate the question with regards to the following ultrapower construction. Assume that $X$ is a set and $(P_{n})_{n}$ is a sequence of partitions of $X$ such that $P_{n+1}$ refines $P_{n}$ for all $n$ and whenever $x,y\in X,x\neq y$ there is some $n$ where $x,y$ belong to different blocks of the partition $P_{n}$. Furthermore, assume that whenever $R_{n}\in P_{n}$ and $R_{n+1}\subseteq R_{n}$ for all $n$ then $\bigcap_{n}R_{n}$ is non-empty. Let $B=\bigcup_{n\in\omega}\{\bigcup\mathcal{R}|\mathcal{R}\subseteq P_{n}\}$. Then $B$ is a Boolean algebra. Let $U$ be an ultrafilter on $B$. We shall use the ultrafilter $U$ to construct an ultrapower.
Suppose $\mathcal{A}$ is a structure. Then let $\mathcal{A}^{(P_{n})_{n}}$ be the collection of all functions $f:X\rightarrow\mathcal{A}$ where there is some natural number $n$ where $P_{n}\preceq\{f^{-1}[\{a\}]|a\in\mathcal{A}\}$ (the ordering $\preceq$ is refinement of partitions). Now let $\simeq_{U}$ be the equivalence relation on $\mathcal{A}^{(P_{n})_{n}}$ where $f\simeq_{U}g$ if and only if $\{x\in X|f(x)=g(x)\}\in U$. One can define the fundamental operations, constants, and relations on $\mathcal{A}^{(P_{n})_{n}}/\simeq_{U}$ as one does with the classical ultrapower construction. Let the structure $\mathcal{A}^{(P_{n})_{n}}/\simeq_{U}$ be denoted by $\mathcal{A}^{(P_{n})_{n}}/U$.
Is there a fairly simple combinatorial characterization of when the ultrapower $V^{(P_{n})_{n}}/U$ well-founded? For instance, in order for $V^{(P_{n})_{n}}/U$ to be well-founded, the ultrafilters $\{\mathcal{R}\subseteq P_{n}|\bigcup\mathcal{R}\in U\}$ must be $\sigma$-complete. On the other hand, if whenever $R_{n}\in\mathcal{U}$ for all $n$, we have $\bigcap_{n}R_{n}\neq\emptyset$, then the ultrapower $V^{(P_{n})_{n}}/U$ is well-founded. Even though I just listed necessary conditions and sufficient conditions for when an ultrapower is well-founded, I do not know of any conditions that are both necessary and sufficient. Can someone give a reference or a proof of combinatorial necessary and sufficient conditions for when such an ultrapower is well-founded? If $V^{(P_{n})_{n}}/U$ is well-founded, then do we necessarily have $\bigcap_{n}R_{n}\neq\emptyset$ whenever $R_{n}\in U$ for all $n$?
$\textrm{This last question was answered affirmatively by Joel David Hamkins}$
$\textbf{Remark}$
Although I phrased this question in terms of a direct limit of a countable sequence of ultrapowers, this characterization also holds for any kind of direct limit of ultrapowers since any possible infinite descending sequence in an ultrapower of a well-founded set must be in a countable direct limit of ultrapowers. Furthermore, the condition that $\bigcap_{n}R_{n}\neq\emptyset$ whenever $R_{n}\in P_{n},R_{n+1}\subseteq R_{n}$ for all $ n$ does not restrict the types of ultrapowers that one can form since one can always extend the set $X$ to a larger set $\hat{X}$ so that $\bigcap_{n}R_{n}\neq\emptyset$ whenever $R_{n}\in P_{n},R_{n+1}\subseteq R_{n}$. For instance, give $X$ the uniformity where the uniform covers are generated by the uniform partitions $P_{n}$. Then the appropriate set $\hat{X}$ is the completion of the uniform space $X$. Equivalently, one can let $\hat{X}=\varprojlim_{n}P_{n}.$ By replacing $X$ with $\hat{X}$ one obtains an isomorphic ultrapower but where $\bigcap_{n}R_{n}\neq\emptyset$ whenever $R_{n}\in P_{n},R_{n+1}\subseteq R_{n}$ for all $n$.