  We present an \textit{ab-initio} disordered local moment theory for the Gibbs
free energy of a magnetic material. Two central objects are calculated: the
lattice Fourier transform of the direct local moment - local moment correlation
functions in the paramagnetic state and local internal magnetic fields as
functions of magnetic order. We identify the potentially most stable magnetic
phases from the first, which can include non-collinear and long-period states
in complex multi-atom unit cells, and extract higher order correlations among
the local moments from the second. We propose that these latter entities
produce a picture of effective multi-site magnetic interactions depending on
the state and extent of magnetic order and discuss its relation to other
approaches. We show how magnetic phase diagrams for temperature, magnetic
field, and lattice structure and also magnetocaloric and mechanocaloric effects
can be obtained from this approach. The theory accurately predicts the order of
transitions and quantifies contributions to first-order and order-order
magnetic phase transitions from both purely electronic sources and
magnetoelastic effects. Our case study is the apparently frustrated magnetism
of the Mn$_3$A class of materials in all its cubic, hexagonal, and tetragonal
structures. The theory produces magnetic phases and transition temperatures in
good agreement with experiment. We explain the first-order triangular
antiferromagnetic to collinear antiferromagnetic transition in cubic Mn$_3$Pt
as a magnetovolume driven effect, and its absence for A=Ir and Rh. We also
construct the magnetic phase diagram of Mn$_3$Pt and explore its potential as a
barocaloric material. Finally, we prepare the groundwork for future fully
relativistic studies of the temperature dependence of the magnetism of Mn$_3$A,
including Mn$_3$Sn, Mn$_3$Ga, and Mn$_3$Ge.
