We continue our investigations on a class of growth models
introduced in a previous paper. Given a nonnegative function $v: \mathbb{Z}^d
\to \mathbb{Z}$ with $v(0) = 0$, we define the space of configurations $\Gamma$
to consist of functions $h: \mathbb{Z}^d \to \mathbb{Z}$ such that $h(i) - h(j)
\leq v(i - j)$ for all i$i, j \epsilon \mathbb{Z}^d$. We then take two
sequances of independent Poisson clocks $(p^{\pm}(i, t): i \epsilon
\mathbb{Z}^d)$ of rates $\lambda^{\pm}$. We start with a possibly random
configuration $h \epsilon \Gamma$. The function $h$ increases (respectively,
decreases) by one unit at site $i$, when the clock $p^{+}(i, \cdot)$
[respectively, $p^{-}(i, \cdot)$] rings and the resulting configuration is
still in $\Gamma$. Otherwise the change in h is suppressed. In this way we have
a process $h(i, t)$ that after a rescaling $u^{\varepsilon}(x, t) =
\varepsilonh([\frac{x}{\varepsilon}], \frac{t}{\varepsilon})$ is expected to
converge to a function $u(x, t)$ that solves a Hamilton-Jacobi equation of the
form $u_t + H(u_x) = 0$. We established this when $\lambda^{-}$ or $\lambda^{+}
= 0$ in the previous paper, employing a strong monotonicity property of the
process $h(i, t)$. Such property is no longer available when both $\lambda^{+},
\lambda^{-}$ are nonzero. In this paper we initiate a new approach to treat the
problem when th dimension is 1 and the set $\Gamma$ can be described by local
constraints on the configuration $h$. In higher dimensions, we can only show
that any limit point of th processes $u^{varepsilon}$ is a process $u$ that
satisfies a Hamilton-Jacobi equation for a suitable (possibly random)
Hamiltonian $H$.