Descendability and descent in topological weaves

We prove a criterion for a finitely presented surjection of algebraic spaces to be descendable in a topological weave. We apply this to show that rational motivic sheaves satisfy v-descent, and the same for \'etale motivic spectra on noetherian finite-dimensional schemes with residue fields of uniformly bounded \'etale cohomological dimension. We also construct the "forgetting supports" isomorphism $f_! \simeq f_*$ for a proper DM morphism of Artin stacks, in rational motivic sheaves.