  We study the off-equilibrium behavior of systems with short-range
interactions driven across a thermal first-order transition, where the dynamics
is exponentially slow. We consider a dynamics that starts in the high-T phase
at time t = t_i<0 and ends in in the low-T phase at t=t_f>0, with a
time-dependent temperature T(t)/T_c \approx 1 - t/t_s, where t_s is the
protocol time scale. A general off-equilibrium scaling (OS) behavior emerges in
the limit of large t_s. We check it at the first-order transition of the
two-dimensional q-state Potts model with q=10 and 20. The numerical results
show evidence of a dynamic transition, where the OS functions show a
spinodal-like singularity. Therefore, the general mean-field picture valid for
systems with long-range interactions is qualitatively recovered, provided the
time dependence is appropriately (logarithmically) rescaled.
