Would dominions fight each other inside a world Commonwealth?

Would a global British Empire have internal conflicts between its dominions?

Political disputes are expected, while separate wars are considered less likely because local forces would belong to a common regional and global command structure.

The answer expects disputes between dominions to take political form within the imagined world Commonwealth. Local armed forces might still exist, but they would sit within regional and global command structures. The dominions would therefore lack the independently organized forces assumed in a war between sovereign states. This is a conditional argument about the proposed command system, not a guarantee that federal states are free from internal violence.