Suppose you are assigning values in $S$ (assume $|S|<\infty$) to nodes of a (directed) graph in a stochastic way. At the beginning, none of the node is assigned values. At the $i^{th}$ step, you (uniformly) randomly pick up a node, say node $\pi(i)$, from those that are not assigned value yet, and assigned it a value $X_i$ according to some probability measure of space $S$, $v_{i/N,d'}(\cdot|G^{N,S,i/N}_{d',pi(i)})$ . Where the probability measure depends on nodes of distance (distance on graph) no larger than $d'$ to $i$ and their temperal (up to step $i$) states. This information is denoted as $G^{N,S,i/N}_{d',\pi(i)}$, N is the size of the whole graph.
I want to prove, roughly speaking, that the pattern (defined latter) induced by $v_{\cdot,d'}(\cdot|\cdot)$ converge in law (to a fixed function) as the graph size tends to infinity and if the pattern of the sequence of graph converges (say a sequence of binary tree, a sequence of 2-dimension square lattice graph).
More speciffically, for a graph G of size N, let d-pattern of graph be
$$L^G_d = \frac{1}{N}\sum\limits_{i=1}^N \delta_{G_{d,i}}$$ where $G_{d,i}$ dentoe the subgraph centered at $i$ consisting of nodes of distance to node $i$ no larger than $d$. $\delta$ is Dirac measure. By definition, for a given graph of radius no larger than $d$, $G_d$, $L^G_d(G_d) = |\{i\in G: G_{d,i} \text{ is isomorphic to } G_d\}|/|G|$
Similarly, for a partially stated graph $G^S$ (a graph with \textbf{some} nodes assigned values), let pattern of (partially ) stated graph be, $$ L^{G,S}_d=\frac{1}{N}\sum\limits_{i=1}^N \delta_{G^S_{d,i}} $$
Back to the previous assignning value to graph process, during the process as more and more nodes are assigned values, the stated graph pattern evolves, denoted by $L^{G^N,S,v_{\cdot,d'},t}_d$, where $\mathcal{G}^S_d$ is the space of partially stated graph of size no larger than $d$. $L^{G^N,S,v_{\cdot,d'},t}_d$ is a $\mathcal{P}(\mathcal{G}^S_d)$ valued stochastic process indexed on $t\in[0,1]$, for $s\in [0,1]$, $L^{G^N,S,v_{\cdot,d'},s}_d$ is the stated graph up to step $[sN]$.
What I want to prove is that, for a sequence a graph $G^N$, if for some $L$, $$ \lim\limits_{N\rightarrow \infty} L^{G^N}\rightarrow^{in\ law} L$$ then for any $v_{\cdot,d'}$, there exists some $L^{\infty,S,v_{\cdot,d'},\cdot}_d$, which is a function from $t$ to $\mathcal{P}(\mathcal{G}^S_d)$ such that $$ \lim\limits_{N\rightarrow \infty}L^{G^N,S,v_{\cdot,d'},\cdot}_d\rightarrow^{in\ law } L^{\infty,S,v_{\cdot,d'},\cdot}_d$$.
Someone must have obtained such results, wo any references are very welcome.