We have studied transport characteristics of mesoscopic multiple-mode superconducting contacts formed between two grains in bulk two-gap magnesium diboride. The experimental setup was realized by driving a normal-metal tip into MgB₂ polycrystalhne sample and proved to be extremely stable, providing possibility to perform pressure experiments at low temperatures. It is argued that in our procedure a small piece of the superconducting electrode is captured by the tip apex and, as a result, two junctions in series are formed: a junction between a lip and MgB₂ grain and a mesoscopic disordered contact between two superconducting pellets. Although the relative weight of the first junction resistance was considerably less, its contribution is shown to be important for the comparison of measured data with expected gap values. Two hallmarks of multiple Andreev reflections inside the MgB₂ - c - MgB₂ contact (c stands for a high-transparent constriction), a zero-bias <$E 1 "/" sqrt |V|>-like singularity of the dc differential conductance and peaks connected to the two gap values, have been revealed. Finally, we report results of a hydrostatic compression experiment showing the evolution of the MgB₂ gap values with pressure. In contrast to the theoretical expectations, we have observed an increase of the smaller gap DELTA_pi whereas the larger gap <$E delta sub sigma> decreased with increasing pressure as it should be for the electron-phonon pairing mechanism. We argue that the so-called separable model of anisotropy effects is insufficient to describe such changes and only improved two-band versions are capable to reproduce the pressure effect on the energy gaps in magnesium diboride.

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