  We expand our study on cubic BiFeO$_3$ alloys presented in [K. Koumpouras and
I. Galanakis, \textit{J. Magn. Magn. Mater} 323, 2328 (2011)] to include also
the BiMnO$_3$ and Bi$_2$MnFeO$_6$ alloys. For the latter we considered three
different cases of distribution of the Fe-Mn atoms in the lattice and six
possible magnetic configurations. We show that Fe and Mn atoms in all cases
under study retain a large spin magnetic moment, the magnitude of which exceeds
the 3 $\mu_B$. Their electronic and magnetic properties are similar to the ones
in the parent BiMnO$_3$ and BiFeO$_3$ compounds. Thus oxygen atoms which are
the nearest-neighbors of Fe(Mn) atoms play a crucial role since they mediate
the magnetic interactions between the transition metal atoms and screen any
change in their environment. Finally, we study the effect of lattice
contraction on the magnetic properties of Bi$_2$MnFeO$_6$.
