  We investigate the proximity effect in junctions between $N=3$
superconductors under commensurate voltage bias. The bias is chosen to
highlight the role of transport processes that exchange multiple Cooper pairs
coherently between more than two superconductors. Such non-local processes can
be studied in the dc response, where local transport processes do not
contribute. We focus on the proximity-induced normal density of states that we
investigate in a wide parameter space. We reveal the presence of deep and
highly tunable pseudogaps and other rich structures. These are due to a static
proximity effect that is absent for $N=2$ and is sensitive to an emergent
superconducting phase associated to non-local coherent transport. In comparison
with results for $N=2$, we find similarities in the signature peaks of multiple
Andreev reflections. We discuss the effect of electron-hole decoherence and of
various types of junction asymmetries. Our predictions can be investigated
experimentally using tunneling spectroscopy.
