  In a previous work, Hoang and Tram discovered a new mechanism for destruction
of nanoparticles due to suprathermal rotation of grains in stationary C-shocks,
which is termed rotational disruption. In this paper, we extend our previous
study for non-stationary shocks driven by outflows and young supernovae
remnants that have dynamical ages shorter than the time required to establish a
stationary C-shock, which is composed of a C-shock and a J-shock tail (referred
as CJ-shock). For the C-shock component, we find that smallest nanoparticles
(size $\lesssim 1$ nm) of weak materials (i.e., tensile strength $S_{\rm max}
\lesssim 10^{9}\ \rm erg\ cm^{-3}$) can be rotationally disrupted due to
suprathermal rotation induced by supersonic neutral drift. For the J-shock
component, although nanoparticles are rotating thermally, the smallest ones can
still be disrupted because the gas is heated to higher temperatures by
J-shocks. We then model microwave emission from rapidly spinning nanoparticles
where the grain size distribution has the lower cutoff determined by rotational
disruption for the different shock models. We also calculate the spectral flux
of microwave emission from a shocked region at distance of 100 pc from the
observer for the different gas density, shock age, and shock velocities. We
suggest that microwave emission from spinning dust can be used to trace
nanoparticles and shock velocities in dense molecular outflows. Finally, we
discuss a new way that can release molecules from the nanoparticle surface into
the gas in the shocked regions, which we name rotational desorption.
