  For fixed positive integers $n$ and $k$, the Kneser graph $KG_{n,k}$ has
vertices labeled by $k$-element subsets of $\{1,2,\dots,n\}$ and edges between
disjoint sets. Keeping $k$ fixed and allowing $n$ to grow, one obtains a family
of nested graphs, each of which is acted on by a symmetric group in a way which
is compatible with all of the other actions. In this paper, we provide a
framework for studying families of this kind using the FI-module theory of
Church, Ellenberg, and Farb, and show that this theory has a variety of
asymptotic consequences for such families of graphs. These consequences span a
range of topics including enumeration, concerning counting occurrences of
subgraphs, topology, concerning Hom-complexes and configuration spaces of the
graphs, and algebra, concerning the changing behaviors in the graph spectra.
