I would be interested to know the answer to the above question for the constructible bounded derived category on complex analytic or complex algebraic manifolds (or some other context). A reference would be helpful. 



Does the nearby cycle functor commute with the Verdier duality?
According to Dimca's "Sheaves in topology" Proposition 4.2.10, the (perverse) nearby and vanishing cycle functors commute non-canonically with the Verdier duality functor if you have field coefficients. For a proof he refers to Brylinski's "Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques" which I don't have available.


