  We describe the $(p,q)$ Fock--Carleson measures for weighted Fock--Sobolev
spaces in terms of the objects $(s,t)$-Berezin transforms, averaging functions,
and averaging sequences on the complex space $\mathbb{C}^n$. The main results
show that while these objects may have growth not faster than polynomials to
induce the $(p,q)$ measures for $q\geq p$, they should be of $L^{p/(p-q)}$
integrable against a weight of polynomial growth for $q<p$. As an application,
we characterize the bounded and compact weighted composition operators on the
Fock--Sobolev spaces in terms of certain Berezin type integral transforms on
$\mathbb{C}^n$. We also obtained estimation results for the norms and essential
norms of the operators in terms of the integral transforms. The results
obtained unify and extend a number of other results in the area.
