Let $(Z_t)_{t\geq0}$ be a homogeneous Poisson point process of intensity $\lambda>0$, such that to each event at time $t$, there is some random location $x\in[0,1]$ attached. More specifically, suppose that the locations are i.i.d., distributed as $X$, whose measure $\mu$ is assumed to be dominated by the Lebesgue measure on $[0,1]$, with density $f_\mu(\cdot)$. Then by standard FCLTs we would expect something like $$ \sqrt T\left(\frac{Z_{Tv}(A)}T-v\lambda\mu(A)\right)_{v\in[0,1]}\longrightarrow\left(\int_A\sqrt{\lambda f_\mu(x)}U(v,x)\ \mathrm dx\right)_{v\in[0,1]} $$ to hold for $A\in\mathcal B[0,1]$, where $U(v,x)=\int_0^vW(s,x)\ \mathrm ds$, for an $L^2$ Gaussian white noise on $[0,\infty)\times[0,1]$. Here the convergence is in the Skorokhod topology.
Now my question is: are there any known (F)CLT results for spatial models (beyond this well-behaved example of the Poisson process with i.i.d. locations) which have such a Gaussian white noise limit? Can such results perhaps be derived from more standard FCLTs having Brownian limits?
Any help or reference is much appreciated.