  We derive the single-fluid evolution equations describing a mixture made of a
gas phase and an arbitrary number of dust phases, generalising the approach
developed in Laibe & Price (2014a). A generalisation for continuous dust
distributions as well as analytic approximations for strong drag regimes are
also provided. This formalism lays the foundation for numeri- cal simulations
of dust populations in a wide range of astrophysical systems while avoiding
limitations associated with a multiple-fluid treatment. The usefulness of the
formalism is illustrated on a series of analytical problems, namely the
dustybox, dustyshock and dustywave problems as well as the radial drift of
grains and the streaming instability in protoplanetary discs. We find physical
effects specific to the presence of several dust phases and multiple drag
timescales, including non-monotonic evolution of the differential velocity
between phases and increased efficiency of the linear growth of the streaming
instability. Interestingly, it is found that under certain conditions, large
grains can migrate outwards in protoplanetary discs. This may explain the
presence of small pebbles at several hundreds of astronomical units from their
central star.
