Madhav Nori invented an original approach to mixed motives, in the spirit of Tannakian theory, but based on a specific Nori's Tannakian theorem. Unfortunately his original work, which has been lectured on some summer schools and conferences is not widely available in written form. Maxim Kontsevich based his approach to periods and their appearance in deformation quantization partly on Nori’s insight.
The construction of Nori motives themselves has been generalised to categories over a base S by Arapura and Ivorra. Arapura’s approach is based on constructible sheaves. His categories allow pull-back and push- forward, the latter being a deep result. The same paper also constructs the weight filtration on Nori motives and establishes the equivalence between Nori motives and André pure motives. Ivorra’s approach is based on perverse sheaves. Compatibility under the six functors formalism? is open in his setting [Huber-Stach]
It is known that Nori’s and Ayoub’s Motivic galois groups agree.
The subcategory of 1-motives in Nori motives agrees with Deligne’s 1-motives, and hence also with 1-motives in Voevodsky’s category.
Madhav Nori, TIFR notes on motives, unpublished, pdf
Jonas von Wangenheim, Nori-Motive und Tannaka-Theorie, arxiv/1111.5146
Nori motives over a base S
For more on this topos theory perspective, see