  In this paper, we are concerned with a class of conservative systems
including asymmetric exclusion processes and zero-range processes as examples,
where some particles are initially placed on $N$ positions. A particle jumps
from a position to another at a rate depending on coordinates of these two
positions and numbers of particles on these two positions. We show that the
hydrodynamic limit of our model is driven by a nonlinear function-valued
ordinary differential equation which is consistent with a mean field analysis.
Furthermore, in the case where numbers of particles on all positions are
bounded by $\mathcal{K}<+\infty$, we show that the fluctuation of our model is
driven by a generalized Ornstein-Uhlenbeck process. A crucial step in proofs of
our main results is to show that numbers of particles on different positions
are approximately independent by utilizing a graphical method.
