Using linear response theory we show that, in a quasi-stationary state, the local multiprobe conductance of a mesoscopic system of non-interacting electrons with a time reversal invariant Hamiltonian does not depend on the local shape of the driving self-consistent potential and thus is entirely determined by the asymptotic values of the potential in the leads. In the ballistic limit, the local conductance in the lateral direction exhibits oscillations depending on the occupation of channels. Scattering by a point impurity leads to softening of the quantized global conductance steps. In addition to that for an attractive scattering potential, a dip occurs in each plateau regime the shape of which is calculated for different values of the potential strength. We also investigate the local conductance for both a point scatterer and a finite scattering region.