We derive Bardeen-like expressions for the transmission probabilities between two multi-probe mesoscopic conductors coupled by a weak tunneling contact. We emphasize especially the dual role of a weak coupling contact as a current source and sink and analyze the magnetic field symmetry. In the limit of a point-like tunneling contact the transmission probability becomes a product of local, partial density of states of the two mesoscopic conductors. We present expressions for the partial density of states in terms of functional derivatives of the scattering matrix with respect to the local potential and in terms of wave functions. We discuss voltage measurements and resistance measurements in the transport state of conductors. We illustrate the theory for the simple case of a scatterer in an otherwise perfect wire. In particular, we investigate the development of the Hall-resistance as measured with weak coupling probes.