Equivalent groupoids have Morita equivalent Steinberg algebras

Let G and H be ample groupoids and let R be a commutative unital ring. We show that if G and H are equivalent in the sense of Muhly-Renault-Williams, then the associated Steinberg algebras are Morita equivalent. We deduce that collapsing a "collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path R-algebra, and therefore a number of graphical constructions which yield Morita equivalent C*-algebras also yield Morita equivalent Leavitt path algebras.