  We prove that the parallel transport of a flat $n-1$-gerbe on any given
target space gives rise to an $n$-dimensional extended homotopy quantum field
theory. In case the target space is the classifying space of a finite group, we
provide explicit formulae for this homotopy quantum field theory in terms of
transgression. Moreover, we use the geometric theory of orbifolds to give a
dimension-independent version of twisted and equivariant Dijkgraaf-Witten
models. Finally, we introduce twisted equivariant Dijkgraaf-Witten theories
giving us in the 3-2-1-dimensional case a new class of equivariant modular
tensor categories which can be understood as twisted versions of the
equivariant modular categories constructed by Maier, Nikolaus and Schweigert.
