  We unite two well known generalisations of the Wadge theory. The first one
considers more general reducing functions than the continuous functions in the
classical case, and the second one extends Wadge reducibility from sets (i.e.,
$\{0,1\}$-valued functions) to $Q$-valued functions, for a better quasiorder
$Q$. In this article, we consider more general reducibilities on the $Q$-valued
functions and generalise some results of L. Motto Ros in the first direction
and of T. Kihara and A. Montalb\'an in the second direction: Our main result
states that the structure of the $\mathbf{\Delta}^0_\alpha$-degrees of
$\mathbf{\Delta}^0_{\alpha+\gamma}$-measurable $Q$-valued functions is
isomorphic to the $\mathbf{\Delta}^0_\beta$-degrees of
$\mathbf{\Delta}^0_{\beta+\gamma}$-measurable $Q$-valued functions, and these
are isomorphic to the generalized homomorphism order on the $\gamma$-th
iterated $Q$-labeled forests.
