Is there a good reference on how RSK (and the 3 other variants) interact with crystal operators on the semi-standard tableaux $(P,Q)$ in the image?

That is, we have biwords, $W$ which are in bijection with pairs of semi-standard tableaux $(P,Q)$ under RSK. Now, we act on $P$ or $Q$ with crystal raising/lowering operators $e_i$ and $f_i$. These actions are defined on SSYTs. What happens with $W$? Theorem 2.2 here almost answers my question (although it just references a paper by Lascoux that is a bit hard to parse), but I would like to know if there is a survey/book that goes into depth on this - in particular when considering the other variants of RSK.

I am in particular interested in variant III in Christian's survey. For a quick reference, I describe this variant on my page here, including an example.