  For a proper local embedding between two Deligne--Mumford stacks Y and X, we
find, under certain mild conditions, a new (possibly non-separated)
Deligne--Mumford stack X', with an etale, surjective and universally closed map
to the target X, and whose fiber product with the image of the local embedding
is a finite union of stacks with corresponding etale, surjective and
universally closed maps to Y. Moreover, a natural set of weights on the
substacks of X' allows the construction of a universally closed push-forward,
and thus a comparison between the Chow groups of X' and X. We apply the
construction above to the computation of the Chern classes of a weighted
blow-up along a regular local embedding via deformation to a weighted normal
cone and localization. We describe the stack X' in the case when X is the
moduli space of stable maps with local embeddings at the boundary. We apply the
methods above to find the Chern classes of the stable map spaces.
