  Bounded and compact differences of two composition operators acting from the
weighted Bergman space $A^p_\omega$ to the Lebesgue space $L^q_\nu$, where
$0<q<p<\infty$ and $\omega$ belongs to the class $\mathcal{D}$ of radial
weights satisfying a two-sided doubling condition, are characterized. On the
way to the proofs a new description of $q$-Carleson measures for $A^p_\omega$,
with $p>q$ and $\omega\in\mathcal{D}$, involving pseudohyperbolic discs is
established. This last-mentioned result generalizes the well-known
characterization of $q$-Carleson measures for the classical weighted Bergman
space $A^p_\alpha$ with $-1<\alpha<\infty$ to the setting of doubling weights.
The case $\omega\in\widehat{\mathcal{D}}$ is also briefly discussed and an open
problem concerning this case is posed.
