  We construct and study a motivic lift of a spectral sequence associated to a
stratified scheme, recently discovered by Petersen in the context of mixed
Hodge theory and $\ell$-adic Galois representations. The original spectral
sequence expresses the compactly supported cohomology of an open stratum in
terms of the compactly supported cohomology of the closures of strata and the
combinatorics of the poset underlying the stratification. Some of its special
cases are classical tools in the study of arrangements of subvarieties and
configuration spaces. Our motivic lift lives in the triangulated category of
\'{e}tale motives and takes the shape of a Postnikov system. We describe its
connecting morphisms and study some of its functoriality properties.
