We give an algorithm for removing stackiness from smooth, tame Artin stacks
with abelian stabilisers by repeatedly applying stacky blow-ups. The
construction works over a general base and is functorial with respect to base
change and compositions with gerbes and smooth, stabiliser preserving maps. As
applications, we indicate how the result can be used for destackifying general
Deligne-Mumford stacks in characteristic zero, and to obtain a weak
factorisation theorem for such stacks. Over an arbitrary field, the method can
be used to obtain a functorial algorithm for desingularising varieties with
simplicial toric quotient singularities, without assuming the presence of a
toroidal structure