I am currently looking at *stratified Mukai flop*. Roughly speaking, this is a construction that, starting with a grassmannian $G$ inside a hyperkahler $X$ produces a birational manifold $X^*$ (with a dual grassmannian $G^*$ inside). The easiest example is the standard Mukai flop, namely the case in which the grassmannian $G$ is a projective space $\mathbb{P}^n$.

I was looking for a concrete example of this situation.

In the case $G=\mathbb{P}^n$, I can construct such an example starting from a rational curve $C\cong\mathbb{P}^1$ inside a $K3$ surface $S$. The Hilbert scheme of points then provides $C^{[n]}\cong\mathbb{P}^n\subset S^{[n]}$.

What about a generic grassmannian $G=G(k,n)$ with $k\neq1,n-1$?

Thank you very much for any comments, remarks and answers.