  We analyze the temporal fluctuations of the flow field associated to a
shear-induced transition in a lyotropic lamellar phase: the layering transition
of the onion texture. In the first part of this work [Salmon et al., submitted
to Phys. Rev. E], we have evidenced banded flows at the onset of this
shear-induced transition which are well accounted for by the classical picture
of shear-banding. In the present paper, we focus on the temporal fluctuations
of the flow field recorded in the coexistence domain. These striking dynamics
are very slow (100--1000s) and cannot be due to external mechanical noise.
Using velocimetry coupled to structural measurements, we show that these
fluctuations are due to a motion of the interface separating the two
differently sheared bands. Such a motion seems to be governed by the
fluctuations of $\sigma^\star$, the local stress at the interface between the
two bands. Our results thus provide more evidence for the relevance of the
classical mechanical approach of shear-banding even if the mechanism leading to
the fluctuations of $\sigma^\star$ remains unclear.
