  We prove several K\"unneth formulas in motivic homotopy categories and deduce
a Verdier pairing in these categories following SGA5, which leads to the
characteristic class of a constructible motive, an invariant closely related to
the Euler-Poincar\'e characteristic. We prove an additivity property of the
Verdier pairing using the language of derivators, following the approach of May
and Groth-Ponto-Shulman; using such a result we show that in the presence of a
Chow weight structure, the characteristic class for all constructible motives
is uniquely characterized by proper covariance, additivity along distinguished
triangles, 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.
