In this work we investigate the Weihrauch degree of the problem $mathsf{DS}$
of finding an infinite descending sequence through a given ill-founded linear
order, which is shared by the problem $mathsf{BS}$ of finding a bad sequence
through a given non-well quasi-order. We show that $mathsf{DS}$, despite being
hard to solve (it has computable inputs with no hyperarithmetic solution), is
rather weak in terms of uniform computational strength. To make the latter
precise, we introduce the notion of the deterministic part of a Weihrauch
degree. We then generalize $mathsf{DS}$ and $mathsf{BS}$ by considering
$oldsymbol{Gamma}$-presented orders, where $oldsymbol{Gamma}$ is a Borel
pointclass or $oldsymbol{Delta}^1_1$, $oldsymbol{Sigma}^1_1$,
$oldsymbol{Pi}^1_1$. We study the obtained $mathsf{DS}$-hierarchy and
$mathsf{BS}$-hierarchy of problems in comparison with the (effective) Baire
hierarchy and show that they do not collapse at any finite level.
