For RSK the answer is "well known". You can find the statements neatly arranged in an article by Christian Krattenthaler http://arxiv.org/abs/math/0510676.
I think the right framework for this question is Sergey Fomin's theory of dual graded graphs.
However, I don't think there are many other insertion algorithms where the Greene-Kleitman invariant is known. One is the insertion algorithm for shifted tableaux, and another, easy one is the pair (BinTree, BinWord).
In fact, whenever you have such a Greene-Kleitman invariant and whenever this invariant behaves well with respect to "promotion", you are in a good position to get a result parallel to http://arxiv.org/abs/math/0604140. For the pair (BinTree, BinWord) this is indeed the case (and interesting), but I never managed to write it up due to time constraints...
For Edelman-Greene the story is slightly different I think. If I recall correctly you can say at least a little bit about the shape of the word by staring long enough at the article by Christian Stump and Luis Serrano http://arxiv.org/abs/1009.4690 or myself http://arxiv.org/abs/1009.3919.
The Kleitman Greene invariants for some insertion algorithms (for the standard case, i.e., where the words are permutations) are described in Sergey Fomin's paper "Schensted algorithms for dual graded graphs":
1) Theorem 4.4.4: Young-Fibonacci insertion (due to Tom Roby and Sergey Fomin, perhaps the invariant for Janvier Nzeutchap's algorithm is different).
2) Just below Proposition 4.5.2: Shifted insertion (attributed to Worley and Bruce Sagan, see Richard Stanley's answer for the description in the semistandard case due to Luis Serrano)
3) Proposition 4.6.2: (BinTree, BinWord)-insertion (independently due to Xavier Viennot)
I'd be interested in learning about Kleitman-Greene invariants for other insertion algorithms. In particular, is it known for domino insertion (as described by Marc van Leeuwen, see also this paper by Thomas Lam)