  Bounded and compact generalized weighted composition operators acting from
the weighted Bergman space $A^p_\omega$, where $0<p<\infty$ and $\omega$
belongs to the class $\mathcal{D}$ of radial weights satisfying a two-sided
doubling condition, to a Lebesgue space $L^q_\nu$ are characterized. On the way
to the proofs a new embedding theorem on weighted Bergman spaces $A^p_\omega$
is established. This last-mentioned result generalizes the well-known
characterization of the boundedness of the differentiation operator
$D^n(f)=f^{(n)}$ from the classical weighted Bergman space $A^p_\alpha$ to the
Lebesgue space $L^q_\mu$, induced by a positive Borel measure $\mu$, to the
setting of doubling weights.
