  It was recently argued that one-dimensional systems of several strongly
interacting fermions of different mass undergo critical transitions between
different spatial orderings when the external confinement adiabatically changes
its shape. In this work, we explore their dynamical properties when finite-time
drivings are considered. By detailed analysis of many-body spectra, we show
that the dynamics is typically guided only by the lowest eigenstates and may be
well-understood in the language of the generalized Landau-Zener mechanism. In
this way, we can capture precisely the dynamical response of the system to the
external driving. As consequence, we show that by appropriate tailoring
parameters of the driving one can target desired many-body state in a finite
time.
