  In this article, we generalize to the case of measured quantum groupoids on a
finite basis some important results concerning equivariant Kasparov theory for
actions of locally compact quantum groups [S. Baaj and G. Skandalis, 1989,
1993]. To every pair $(A,B)$ of C*-algebras continuously acted upon by a
regular measured quantum groupoid on a finite basis $\cal G$, we associate a
$\cal G$-equivariant Kasparov theory group ${\sf KK}_{\cal G}(A,B)$. The
Kasparov product generalizes to this setting. By applying recent results
concerning actions of regular measured quantum groupoids on a finite basis [S.
Baaj and J. C., 2015; J. C., 2017], we obtain two canonical homomorphisms
$J_{\cal G}:{\sf KK}_{\cal G}(A,B)\rightarrow{\sf KK}_{\widehat{\cal
G}}(A\rtimes{\cal G},B\rtimes{\cal G})$ and $J_{\widehat{\cal G}}:{\sf
KK}_{\widehat{\cal G}}(A,B)\rightarrow{\sf KK}_{\cal G}(A\rtimes\widehat{\cal
G},B\rtimes\widehat{\cal G})$ inverse of each other through the Morita
equivalence coming from a version of the Takesaki-Takai duality theorem [S.
Baaj and J. C., 2015; J. C., 2017]. 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 [S. Baaj and J. C., 2015].
