I have a user with an alignment that has the stationing labels in 20 and 60 scales annotative. He needs to move one of the 60 scale labels because at that scale it overlaps another label nearbye. However when he makes the 60 scale the active scale and selects the label and moves it with a grip like normal, it also modifies the positioning of the 20 scale label. Not sure if it matters, but unlike normal annotative text, it does not shadow the other scales when he selects it, though that might be an independent setting.
Civil 3D labels do not include the ability to be moved independently at different scales. That would be a nice feature, but it's just not possible at this point in time.
in order to have different label positions at different scales it is necessary to make a seperate label for each scale and put them on different layers and turn them on/off in their respective viewports.