This might be a place to discuss extension of functors on rings to functors on schemes. Devices for such extensions include Thomason’s hypercohomology construction. Another device might be Jouanolou’s trick.
The six functors does this sometimes.
Guillen-Navarro Aznar in IHES paper.
Thomason’s hypercohomology construction, mentioned by Geisser in K-th handbook.
See a note under unramified cohomology.
Gillet and Soule, in the Motives descent K-th paper, shows (according to the review) something general can be said about the possibility of deriving certain functors to obtain cohomology theories on the cat of all varieties (maybe over a field of char zero)
nLab page on B35 Extending cohomology theories to bigger categories