The nature of the ground field $K$ doesn't seem to be very relevant. In general $S$ won't even be noetherian. The only positive result I know is the following: if $S_i$ is regular for all sufficiently large $i$, and the map $S_j\to S_i$ is etale for all sufficiently large $i,j$, then $S$ is noetherian and regular (and $dim S=dim S_i$ for all sufficiently large $i$.

The nature of the ground field $K$ doesn't seem to be very relevant. In general $S$ won't even be noetherian. The only positive result I know is the following: if $S_i$ is regular for all sufficiently large $i$, and the map $S_j\to S_i$ is etale for all sufficiently large $i,j$, then $S$ is noetherian and regular (and $\dim S=\dim S_i$ for all sufficiently large $i$).