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 étale motivic spectra satisfy $h$-descent on noetherian finite-dimensional schemes with residue fields of uniformly bounded étale cohomological dimension. We also show that rational motivic cohomology satisfies arc-descent in weights $\le 1$, and we construct the ``forgetting supports’’ isomorphism $f_{!}\simeq f_{\ast }$ for a proper DM-type morphism of Artin stacks, in rational motivic sheaves.