  We study the $\beta$ analogue of the nonintersecting Poisson random walks. We
derive a stochastic differential equation of the Stieltjes transform of the
empirical measure process, which can be viewed as a dynamical version of the
Nekrasov's equation in [7, Section 4]. We find that the empirical measure
process converges weakly in the space of c\'adl\'ag measure-valued processes to
a deterministic process, characterized by the quantized free convolution, as
introduced in [11]. For suitable initial data, we prove that the rescaled
empirical measure process converges weakly in the space of distributions acting
on analytic test functions to a Gaussian process. The means and the covariances
are universal, and coincide with those of $\beta$-Dyson Brownian motions with
the initial data constructed by the Markov-Krein correspondence. Especially,
the covariance structure can be described in terms of the Gaussian Free Field.
Our proof relies on integrable features of the generators of the
$\beta$-nonintersecting Poisson random walks, the method of characteristics,
and a coupling technique for Poisson random walks.
