  In this paper we extend our approach to modeling multifrequency emission
variability from blazars to include external-radiation Compton (ERC) emission
and electron energy losses from inverse Compton scattering of seed photons
originating outside the jet. We consider seed-photon emission from a dusty
molecular torus and the broad line region (BLR) surrounding the central engine.
We establish constraints on the properties of the molecular torus and BLR under
which synchrotron self-Compton (SSC) emission dominates such that the results
obtained in our previous paper are applicable. The focus of this study is on
relative time delays between the light curves observed at different
frequencies. For definiteness, we consider emission resulting from a collision
between relativistic shocks, but the results apply more generally to conditions
involving acceleration of relativistic electrons and/or magnetic field
amplification at any type of front. Unlike SSC emission, ERC flares involving a
constant field of seed photons are not delayed by light travel time of the seed
photons. The main cause of delays is from radiative energy losses, which result
in frequency stratification behind the front and quench the flare first at the
highest frequencies, progressing to lower frequencies as time advances.
However, if the spectrum of electrons injected at the shock front is
characterized by a relatively high value of the minimum energy (a Lorentz
factor $\gamma_{min}\sim100$ is sufficient), the ERC flare in the X-ray band
can be delayed and may even peak after the injection has ceased. This effect is
strongly frequency dependent, with a longer lag at lower frequencies and an
X-ray spectral index that changes rapidly from positive (inverted spectrum) to
steep values.
