Structural properties of amorphous In-based oxides, In-X-O with X=Zn, Ga, Sn, or Ge, are investigated using ab initio molecular dynamics liquid-quench simulations. The results reveal that indium retains its average coordination of 5.0 upon 20% X fractional substitution for In, whereas X cations satisfy their natural coordination with oxygen atoms. This finding suggests that the carrier generation is primarily governed by In atoms, in accord with the observed carrier concentration in amorphous In-O and In-X-O. At the same time, the presence of X affects the number of six-coordinated In atoms as well as the oxygen sharing between the InO6 polyhedra. Based on the obtained interconnectivity and spatial distribution of the InO6 and XOx polyhedra in amorphous In-X-O, composition-dependent structural models of the amorphous oxides are derived. The results help explain our Hall mobility measurements in In-X-O thin films grown by pulsed-laser deposition and highlight the importance of long-range structural correlations in the formation of amorphous oxides and their transport properties.