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Let's say I have a cube and a sphere, and I make the cube the parent of the sphere. Then I link-duplicate both the cube and sphere with AltD. Then I start moving/rotating/etc. the duplicates around in object mode, together as a unit.

Later, I discover that I want to change the position of the sphere (child) with respect to the cube (parent). I re-position the original sphere, but the duplicate sphere does not change position.

I can manually copy the transform values from the original to the duplicate, but is there a way to automatically have the transform values be linked?

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To duplicate multiple objects in different positions and keep the relationship between each instance, you want to use DupliGroups.

  • Select the objects you want to duplicate.
  • Add them to a group ⎈ CtrlG
  • You can rename the group so that it is more meaningful.
  • Add an empty
  • Set the empties Duplication setting to group and choose the group you created.
  • Repeat the last two steps as needed.

dupligroup settings

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  • $\begingroup$ In Blender 2.8 it's unclear where the "Duplication" setting is $\endgroup$
    – Anson Kao
    Feb 19 '20 at 21:28
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In Blender 2.8+, this has changed a bit from @sambler's answer.

Changes since older versions:

  • Groups are now superseded by something called Collections (see video)
  • The "Duplication" setting in an empty is now superseded by a section called "Instancing".

The updated steps are:

  • Select the objects you want to duplicate.
  • Add them to a new Collection by right clicking or using hotkey M.
  • You can rename the Collection so that it is more meaningful.
  • Add an empty
  • Set the empty's Instancing setting to "Collection" and choose the collection you created.
  • Repeat the last two steps as needed.

Instancing in Blender 2.8+

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