I generally call and make changes right away when I get unacceptable schedule changes from Delta. There's usually more flexibility on the flights the earlier you call, though you do have to balance that out knowing that additional changes are always possible.

Like you, I sometimes make complicated international trips with small children and various family members coming together from different parts of the world. The logistics are never fun, but keeping on top of changes helps. Thank heavens my youngest teen is now 15 and no longer needs the unaccompanied minor service to fly internationally.