We define an action of the k-strand braid group on the set of clusters for Gr(k,n), whenever k divides n. This action preserves the underlying quivers, defining a homomorphism from the braid group to the "cluster modular group" for Gr(k,n), the latter of which is an "SL_k version" of a mapping class group. Then we apply our results to certain Grassmannians of "finite mutation type," proving the Gr(3,9) case of a conjecture of Fomin-Pylyavskyy describing the cluster combinatorics for Gr(3,n) in terms of Kuperberg's basis of non-elliptic webs. We obtain a similar description of the cluster combinatorics for Gr(4,8). Speaker(s): Chris Fraser (IUPUI)