I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?
More generally, what can be said if one allows general schemes over a sufficiently good base $S$ (probably something like geometrically unibranch?) or change the topology (e.g., étale motivic cohomology)?
Still more generally, what can be said if one uses Fulton-Macpherson's bivariant theories or Mixed Weil cohomology theories?