  We develop a theory of $\times$-homotopy, fundamental groupoids and covering
spaces that apply to non-simple graphs, generalizing existing results for
simple graphs. We prove that $\times$-homotopies from finite graphs can be
decomposed into moves which adjust at most one vertex at a time, generalizing
the spider lemma of \cite{CS1}. We define a notion of homotopy covering map and
develop a theory of universal covers and deck transformations, generalizing
\cites{TardifWroncha, Matsushita} to non-simple graphs. We examine the case of
reflexive graphs, where each vertex has at least one loop. We also prove that
these homotopy covering maps satisfy a homotopy lifting property for arbitrary
graph homomorphisms, generalizing path lifting results of \cites{Matsushita,
TardifWroncha}.
