  Polarimetric measurements, especially if extended at high energy, are
expected to provide important insights into the mechanisms underlying the
acceleration of relativistic particles in jets. In a previous work we have
shown that the polarization of the synchrotron X-ray emission produced by
highly energetic electrons accelerated by a mildly relativistic shock carries
essential imprints of the geometry and the structure of the magnetic fields in
the downstream region. Here we present the extension of our analysis to the
non-stationary case, especially suitable to model the highly variable emission
of high-energy emitting BL Lacs. We anticipate a large ($\Pi \approx 40\%$),
almost time-independent degree of polarization in the hard/medium X-ray band, a
prediction soon testable with the upcoming mission {\it IXPE}. The situation in
other bands, in particular in the optical, is more complex. A monotonic
decrease of the optical degree of polarization is observed during the
development of a flare. At later stages $\Pi$ reaches zero and then it starts
to increase, recovering large values at late times. The instant at which
$\Pi=0$ is marked by a rotation of the polarization angle by $90$ degrees.
However, at optical frequencies it is likely that more than one region
contributes to the observed emission, potentially making it difficult to detect
the predicted behavior.
