The answer is motivic stable homotopy theory! This assigns to any reasonable scheme $S$ a triangulated category $SH(X)$ whose objects represent generalized cohomology theories for $S$-schemes, and there is also a similar category $DM(S)$ whose objects represent ordinary cohomology theories.

These theories do almost everything you ask for, in particular they give a nice framework for cohomology and homology including Borel-Moore and compact support versions.

I could say a lot about this, but I have to go to sleep. Let me just point what is probably the best reference, namely the introduction to this preprint of Déglise and Cisinski:

http://www.math.univ-toulouse.fr/~dcisinsk/DM.pdf

The answer is motivic stable homotopy theory! This assigns to any reasonable scheme $S$ a triangulated category $SH(S)$ whose objects represent generalized cohomology theories for $S$-schemes, and there is also a similar category $DM(S)$ whose objects represent ordinary cohomology theories.

These theories do almost everything you ask for, in particular they give a nice framework for cohomology and homology including Borel-Moore and compact support versions.

I could say a lot about this, but I have to go to sleep. Let me just point what is probably the best reference, namely the introduction to this preprint of Déglise and Cisinski:

http://www.math.univ-toulouse.fr/~dcisinsk/DM.pdf