[FOM] Question on ultraproducts

[Artyom Chernikov]

Suppose $A_i, i \in \omega$ is an indexed family of structures, S is a
permutation of naturals and U is an ultrafilter on $\omega$.
Obviously $\prod_{ i\in\omega }A_i/U$ is not isomorphic to $\prod_{
i\in\omega }A_{S(i)}/U$ in general.
Can we somehow specify (without complete trivialization) type of U, or
S, or maybe complexity of A_i to keep such ultraproducts isomorphic. I
mean to specify respective parameters in mutually independent fashion,
not along lines of "S has infinite set of fixed points, wich is in U".

Thanks in advance,
Artyom Chernikov