  We study damage spreading in models of two-dimensional systems undergoing
first order phase transitions. We consider several models from the same
non-conserved order parameter universality class, and find unexpected
differences between them. An exact solution of the Ohta-Jasnow-Kawasaki model
yields the damage growth law $D \sim t^{\phi}$, where $\phi = t^{d/4}$ in $d$
dimensions. In contrast, time-dependent Ginzburg-Landau simulations and Ising
simulations in $d= 2$ using heat-bath dynamics show power-law growth, but with
an exponent of approximately $0.36$, independent of the system sizes studied.
  In marked contrast, Metropolis dynamics shows damage growing via $\phi \sim
1$, although the damage difference grows as $t^{0.4}$. PACS: 64.60.-i, 05.50.+q
