Let P be a transition operator over a countable set which is invariant under the action of a locally compact group G with compact point stabilizers. We give upper bounds for the norm and spectral radius of P acting on \ell^s(X,\mu), where 1 < s < infinity and \mu is a measure on X satisfying a compatibility condition with respect to G. When G is amenable, our inequalities become equalities involving the modular function of G. When G, besides being amenable, acts with finitely many orbits then this allows easy computation of norms and spectral radii via reduction to a finite matrix. For unimodular groups there are further simplifications. A variety of examples is given, including the (linear) buildings of type \tilde A_{n-1} associated with PGL(n,F) over a local field F. These results extend previous work of Soardi and Woess, Salvatori, and Saloff-Coste and Woess, where only reversible Markov operators and the case s=2 were studied.