  We study the diffusive dynamics of the classical one-dimensional lattice
model of mobile particles featuring the incoherent metallic state of the
organic TTF-CA found in the vicinity of the neutral-ionic(NI)-transition. The
particles are strongly correlated and feel the alternating site potentials,
exhibiting uniform ionic (I) Mott insulating and neutral band insulating (N)
phases when the Coulomb interaction and site potentials are large,
respectively. We focus on the neutral-ionic domain walls (NIDW) activated by
their competition, which is typically regarded as fractionalized particles. The
finite temperature phase diagram reveals a thermodynamically stable NIDW phase
not found in previous literature that emerges at the triple point, where the
transition line splits into two first-order lines toward the critical
endpoints. We analyze the long-time behavior of dynamics of the NIDWs and find
that it captures the incoherent transport of the system. The conductivity
derived from the diffusion constant and the number population of NIDWs shows a
strong enhancement inside the NIDW phase, which can explain the large
conductivity at the NI crossover recently found in TTF-CA.
