  In this article, we study the forward dynamical behavior of nonautonomous
lattice systems. We first construct a family of sets
$\{\mathcal{A}_\varepsilon(\sigma)\}_{\sigma\in \Sigma}$ in arbitrary small
neighborhood of a global attractor of the skew-product flow generated by a
general nonautonomous lattice system, which is forward invariant and uniformly
forward attracts any bounded subset of the phase space. Moreover, under some
suitable conditions, we further construct a family of sets
$\{\mathcal{B}_\varepsilon(\sigma)\}_{\sigma\in \Sigma}$ such that it uniformly
forward exponentially attracts bounded subsets of the phase space. As an
application, we study the discrete Gray-Scott model in detail and illustrate
how to apply our abstract results to some concrete lattice system.
