Great suggestion by Spaceman,
but you might get the same problem, it probably has to do with differing sizes of your different layers.
The Anchorpoints will be somewhere else, and that causes the shift.
Also, it’s not evident to copy position or rotations from a parented layer to a different layer, because the actual keyframed value doesn’t describe exactly where the layer is, because it is in relation to the parent..
8-P Hope that’s clear.
Anyway, use Spaceman’s suggestion, then move the anchorpoint around untill it fits again.