Lifting (co)stratifications between tensor triangulated categories

We give necessary and sufficient conditions for stratification and costratification to descend along a coproduct preserving, tensor-exact functor between tensor-triangulated categories which are rigidly-compactly generated by their tensor units. We then apply these results to non-positive commutative DG-rings and connective ring spectra. In particular, this gives a support-theoretic classification of (co)localizing subcategories, and thick subcategories of compact objects of the derived category of a non-positive commutative DG-ring with finite amplitude. Our results also establish the telescope conjecture in this setting. For a non-positive commutative DG-ring $A$, we also investigate whether certain finiteness conditions in $\mathsf{D}(A)$ (for example, proxy-smallness) can be reduced to questions in the better understood category $\mathsf{D}(H^0A)$. For a connective commutative ring spectrum $R$ we give a necessary and sufficient condition for $\mathsf{D}(R)$ to be stratified and costratified by $\pi_0R$.