  In this paper we are concerned with hydrodynamics of a class of $N$-urn
linear systems, which include voter models, pair-symmetric exclusion processes
and binary contact path processes on $N$ urns as special cases. We show that
the hydrodynamic limit of our process is driven by a
$\left(C[0,1]\right)^\prime$-valued linear ordinary differential equation and
the fluctuation of our process, i.e, central limit theorem from the
hydrodynamic limit, is driven by a $\left(C[0, 1]\right)^\prime$-valued
Ornstein-Uhlenbeck process. To derive above main results, we need several
replacement lemmas. An extension in linear systems of Chapman-Kolmogorov
equation plays key role in proofs of these replacement lemmas.
