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].