In this article, we generalize to the case of measured quantum groupoids on a finite basis some important results concerning the equivariant Kasparov theory for actions of locally compact quantum groups, see Baaj and Skandalis (1989, 1993). To every pair $(A,B)$ of $\\mathrm{C}^\*$-algebras continuously acted upon by a regular measured quantum groupoid on a finite basis $\\mathcal{G}$, we associate a $\\mathcal{G}$-equivariant Kasparov theory group $\\mathsf{KK}{\\mathcal{G}}(A,B)$. The Kasparov product generalizes to this setting. By applying recent results by Baaj and Crespo (2017, 2018) concerning actions of regular measured quantum groupoids on a finite basis, we obtain two canonical homomorphisms $J{\\mathcal{G}}:\\mathsf{KK}{\\mathcal{G}}(A,B)\\rightarrow\\mathsf{KK}{\\hat{\\mathcal{G}}}(A\\rtimes{\\mathcal{G}},B\\rtimes{\\mathcal{G}})$ and $J{\\hat{\\mathcal{G}}}:\\mathsf{KK}{\\hat{\\mathcal{G}}}(A,B)\\rightarrow\\mathsf{KK}\_{\\mathcal{G}}(A\\rtimes\\hat{\\mathcal{G}},B\\rtimes\\hat{\\mathcal{G}})$ inverse of each other through the Morita equivalence coming from a version of the Takesaki–Takai duality theorem. We investigate in detail the case of colinking measured quantum groupoids. In particular, if $\\mathbb{G}\_1$ and $\\mathbb{G}\_2$ are two monoidally equivalent regular locally compact quantum groups, we obtain a new proof of the canonical equivalence of the associated equivariant Kasparov categories, see Baaj and Crespo (2017).