  We discuss the large-scale geometry of pure mapping class groups of locally
finite, infinite graphs, motivated by recent work of Algom-Kfir--Bestvina and
the work of Mann--Rafi on the large-scale geometry of mapping class groups of
infinite-type surfaces. Using the framework of Rosendal for coarse geometry of
non-locally compact groups, we classify when the pure mapping class group of a
locally finite, infinite graph is globally coarsely bounded (an analog of
compact) and when it is locally coarsely bounded (an analog of locally
compact). Our techniques give lower bounds on the first integral cohomology of
the pure mapping class group for some graphs and allow us to compute the
asymptotic dimension of all locally coarsely bounded pure mapping class groups
of infinite rank graphs. This dimension is always either zero or infinite.
