I understand why this doesn't work - the green box doesn't directly inherit the transforms of the path so it doesn't translate the transforms to the child object (axis stays at origin through the whole path movement), but want to know if there's another way of getting it to work! :) This is the basics of how the scene is…
Found a solution - assigned controller in the motion tab instead of path deform modifier and then assigned it to the path - child dummy now follows the parent. :)