  We study off-equilibrium behaviors at first-order transitions (FOTs) driven
by a time dependence of the temperature across the transition point Tc, such as
the linear behavior T(t)/Tc = 1 - t/ts where ts is a time scale. In particular,
we investigate the possibility of nontrivial off-equilibrium scaling behaviors
in the regime of slow changes, corresponding to large ts, analogous to those
arising at continuous transitions, which lead to the so-called Kibble-Zurek
mechanism.
  We consider the 2D Potts models which provide an ideal theoretical laboratory
to investigate issues related to FOTs driven by thermal fluctuations. We put
forward general ansatzes for off-equilibrium scaling behaviors around the time
t=0 corresponding to Tc. Then we present numerical results for the q=10 and
q=20 Potts models. We show that phenomena analogous to the Kibble-Zurek
off-equilibrium scaling emerge also at FOTs with relaxational dynamics, when
appropriate boundary conditions are considered, such as mixed boundary
conditions favoring different phases at the opposite sides of the system, which
enforce an interface in the system.
