  In this paper we are concerned with a generalized $N$-urn Ehrenfest model,
where balls keeps independent random walks between $N$ boxes uniformly laid on
$[0, 1]$. After a proper scaling of the transition rates function of the
aforesaid random walk, we derive the hydrodynamic limit of the model, i.e., the
law of large numbers which the empirical measure of the model follows, under an
assumption where the initial number of balls in each box independently follows
a Poisson distribution. We show that the empirical measure of the model
converges weakly to a deterministic measure with density driven by an integral
equation. Furthermore, we derive non-equilibrium fluctuation of the model, i.e,
the central limit theorem from the above hydrodynamic limit. We show that the
non-equilibrium fluctuation of the model is driven by a measure-valued
time-inhomogeneous generalized O-U process. At last, we prove a large deviation
principle from the hydrodynamic limit under an assumption where the transition
rates function from $[0, 1]\times [0, 1]$ to $[0, +\infty)$ of the aforesaid
random walk is a product of two marginal functions from $[0, 1]$ to $[0,
+\infty)$.
