  Let $G$ and $H$ be Hausdorff 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 of locally
constant $R$-valued functions with compact support 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.
