  We study large random partitions boxed into a rectangle and coming from skew
Howe duality, or alternatively from dual Schur measures. As the sides of the
rectangle go to infinity, we obtain: 1) limit shape results for the profiles
generalizing the Vershik--Kerov--Logan--Shepp curve; and 2) universal edge
asymptotic results for the first parts in the form of the Tracy--Widom
distribution, as well as less-universal critical regime results introduced by
Gravner, Tracy and Widom. We do this for a large class of Schur parameters
going beyond the Plancherel or principal specializations previously studied in
the literature, parametrized by two real valued functions $f$ and $g$.
Connections to a Bernoulli model of (last passage) percolation are explored.
