  The innermost parts of powerful jets in active galactic nuclei are surrounded
by dense, high-velocity clouds from the broad-line region, which may penetrate
into the jet and lead to the formation of a strong shock. Such jet-cloud
interactions are expected to have measurable effects on the $\gamma$-ray
emission from blazars. We characterise the dynamics of a typical cloud-jet
interaction scenario, and the evolution of its radiative output in the
0.1-30~GeV energy range, to assess to what extent these interactions can
contribute to the $\gamma$-ray emission in blazars. We use semi-analytical
descriptions of the jet-cloud dynamics, taking into account the expansion of
the cloud inside the jet and its acceleration. Assuming that electrons are
accelerated in the interaction and making use of the hydrodynamical
information, we then compute the high-energy radiation from the cloud,
including the absorption of $\gamma$-rays in the ambient photon field through
pair creation. Jet-cloud interactions can lead to significant $\gamma$-ray
fluxes in blazars with a broad-line region, in particular when the cloud
expansion and acceleration inside the jet are taken into account. This is
caused by 1) the increased shocked area in the jet, which leads to an increase
in the energy budget for the non-thermal emission; 2) a more efficient inverse
Compton cooling with the boosted photon field of the broad-line region; and 3)
an increased observer luminosity due to Doppler boosting effects. For typical
broad-line region parameters, either (i) jet-cloud interactions contribute
significantly to the persistent $\gamma$-ray emission from blazars or (ii) the
broad-line region is far from spherical or the fraction of energy deposited in
non-thermal electrons is small.
