This notion has arisen in various contexts. In particular, the case of trees was studied in detail in an older paper Asymptotic fringe distributions ..... Our interest arises from the particular example of the stochastic mean-field model of distance where the limit PWIT arises from the model of $n$ random points with all n-choose-2 inter-point distances being independent. The recent survey The objective method (with Mike Steele) contains some basic general theory. For instance, local weak convergence implies convergence of minimal spanning trees to the (wired) minimal spanning forest on the limit network. See also Gamarnik et al Maximum Weight Independent Sets and Matchings in Sparse Random Graphs.
Infinite networks which arise as limits of uniformly-rooted finite networks automatically inherit a certain stationarity property, "unimodularity". Surprisingly, it is not obvious that every infinite network with this stationarity property can be constructed as a weak limit of finite networks (in talks in 2004 I claimed to have proved this, but the proof collapsed in 2005). In Processes on unimodular random networks (with Russ Lyons) the class of unimodular networks is studied.
A nice use of local weak convergence is in the study of random quadrangulations: see [Krikun] and [Chassaing-Durhuus].