Locally free actions of groupoids and proper topological correspondences

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Let (G,α) and (H,β) be locally compact Hausdorff groupoids with Haar systems, and let (X,λ) be a topological correspondence from (G,α) to (H,β) which induce the C*-correspondence H(X) C*(G,α)→ C*(H,β). We give sufficient topological conditions which when satisfied the C*-correspondence H(X) is proper, that is, the C*-algebra C*(G,α) acts on the Hilbert C*(H,β)-module H(X) via the comapct operators. Thus a proper topological correspondence produces an element in KK(C*(G,α),C*(H,β)).