  In earlier work, Billey--Konvalinka--Swanson studied the asymptotic
distribution of the coefficients of Stanley's $q$-hook length formula, or
equivalently the major index on standard tableaux of straight shape and certain
skew shapes. We extend those investigations to Stanley's $q$-hook-content
formula related to semistandard tableaux and $q$-hook length formulas of
Bj\"orner--Wachs related to linear extensions of labeled forests. We show that,
while their coefficients are ``generically'' asymptotically normal, there are
uncountably many non-normal limit laws. More precisely, we introduce and
completely describe the compact closure of the metric space of distributions of
these statistics in several regimes. The additional limit distributions involve
generalized uniform sum distributions which are topologically parameterized by
certain decreasing sequence spaces with bounded $2$-norm. The closure of these
distributions in the L\'evy metric gives rise to the space of DUSTPAN
distributions. As an application, we completely classify the limiting
distributions of the size statistic on plane partitions fitting in a box.
