This is a rather imprecise question but i think this could become a interesting pool of ideas and comments.

The theory of motives has evolved to a complex field of research the moment Voevodsky (and others) have introduced the triangulated categories and concepts of homotopy theory to the field. The/one motivation behind (stable) derivators is to enable a functorial cone construction. This is a problem arising in the theory of triangulated categories.

What improvements in the theory of motives could be derived from the theory of derivators?

Noticing that both concepts where introduced by late Grothendieck made me think if maybe there originally was a conceptual link between these topics in the first place, when he came up with the derivators.

schemesover some base, and imposing various axioms. The resulting notion, which he calls analgebraic derivator, essentially captures the whole formalism of six functors. I am not sure, but I believe that Grothendieck probably did not have anything like this in mind when he wrote Pursuing Stacks or Les Derivateurs (and I will be very happy if an expert could confirm). $\endgroup$Plongement de certaines théories homotopiques de Quillen dans les dérivateurs, by Olivier Renaudin $\endgroup$6more comments