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We prove several K\"unneth formulas in motivic homotopy categories and deducea Verdier pairing in these categories following SGA5, which leads to thecharacteristic class of a constructible motive, an invariant closely related tothe Euler-Poincar\'e characteristic. We prove an additivity property of theVerdier pairing using the language of derivators, following the approach of Mayand Groth-Ponto-Shulman; using such a result we show that in the presence of aChow weight structure, the characteristic class for all constructible motivesis uniquely characterized by proper covariance, additivity along distinguishedtriangles, refined Gysin morphisms and Euler classes. In the relative setting,we prove the relative K\"unneth formulas under some transversality conditions,and define the relative characteristic class.

K\"unneth formulas for motives and additivity of traces