The asymptotic behavior of solid closure in mixed characteristic

We study how solid closure in mixed characteristic behaves after taking ultraproducts. The ultraproduct will be chosen so that we land in equal characteristic, and therefore can make a comparison with tight closure. As a corollary we get an asymptotic version of the Hochster-Roberts invariant theorem in dimension three: if $R$ is a mixed characteristic (cyclically) pure 3-dimensional local subring of a regular local ring $S$, then $R$ is Cohen-Macaulay, provided the ramification of $S$ is large with respect to its dimension and residual characteristic, and with respect to the multiplicity of $R$.