Recently, Algom-Kfir and Bestvina introduced mapping class groups of locally finite graphs as a proposed analog of Out(F_n) in the infinite-type setting. In this talk we will introduce the classification of infinite-type graphs, their mapping class groups, and some important types of elements in these groups. Using a framework established by Rosendal, we will then discuss the coarse geometry of the pure mapping class groups and related properties, including results on asymptotic dimension. This is joint work with George Domat and Sanghoon Kwak.