Abstract: For any Lie group G we define moduli spaces resp. spectra D_d^G(X) of proper G-manifolds of dimension d which are properly and equivariantly parametrized by a proper G-space X, generalizing moduli spaces of manifolds introduced by Galatius/Randal-Williams. The resulting spectrum valued functor (G,X) |-> D_d^G(X) defines a locally finite equivariant homology theory which for trivial G and X a point partially recovers the identification of Galatius/Madsen/Tillmann/Weiss resp. GRW of the cobordism category with an infinite loop space of a shifted tangential Thom spectrum. As a further application we find a functor from proper G-spaces to spectra which represents locally finite G-bordism.