  Given two equivalent locally compact Hausdorff groupoids, the Bost conjecture
with Banach algebra coefficients is true for one if and only if it is true for
the other. This also holds for the Bost conjecture with C*-coefficients. To
show these results, the functoriality of Lafforgue's KK-theory for Banach
algebras and groupoids with respect to generalised morphisms of groupoids is
established. It is also shown that equivalent groupoids have Morita equivalent
L^1-algebras (with Banach algebra coefficients).
