I could have sworn that, sometime in the last month, I saw a question along the following lines:
Let $v$ and $w$ be elements of the symmetric group. Let $X(v) \subset \mathrm{Flags}(n)$ be the Schubert variety indexed by $v$. Let $w X(v)$ be the image of $X(v)$ under $S_n \subset GL_n$ acting on $\mathrm{Flags}(n)$. What is the limit of $w X(v)$ under the Gelfand-Tsetlin degeneration?
I don't have any new ideas about this question, but I recently learned that Allen Knutson is interested in some related questions, so I wanted to put him in touch with the OP.
I can't find this question anywhere. Was it deleted? Does anyone remember who posted it?