For low biases the linear conductance of quantum dots is based on elastic transport processes. At finite bias in the Coulomb blockade regime, inelastic cotunneling sets in once the applied bias exceeds the energy between ground and excited state in the dot. Here we report on transport experiments through an Aharonov-Bohm ring containing a quantum dot in each arm of the ring. The tunnel coupling between the two dots can be tuned by electrostatic gates. For strong tunnel coupling and low bias we observe pronounced Aharonov-Bohm oscillations in the ring with visibilities exceeding 80%. For quantum dots which are purely capacitively coupled, the Aharonov-Bohm amplitude is reduced to a more standard 10%. For finite bias, where transport through excited states becomes possible and a conductance onset is observed, the visibility of the Aharonov-Bohm oscillations remains basically unchanged, while the phase typically undergoes a change of pi. We discuss these observations in view of the possible elastic and inelastic transport processes and their contributions to coherent transport.