In this article, we generalize to the case of regular locally compact quantum groups, two important results concerning actions of compact quantum groups (see [?] and [?]). Let $G_1$ and $G_2$ be two monoïdally equivalent regular locally compact quantum groups in the sense of De Commer (see [?], [?]). We introduce an induction procedure and we build an equivalence of the categories $A^{G_1}$ and $A^{G_2}$ consisting of continuous actions of $G_1$ and $G_2$ on C*-algebras. As an application of this result, we derive a canonical equivalence of the categories $KK^{G_1}$ and $KK^{G_2}$. We introduce and investigate a notion of actions on C*-algebras of measured quantum groupoids (see [?]) on a finite basis. The proof of the equivalence between $KK^{G_1}$ and $KK^{G_2}$ relies on a version of the Takesaki-Takai duality theorem for continuous actions on C*-algebras of measured quantum groupoids on a finite basis.