It's a tad difficult to be precise without some sample images.
If the objects are not identical....
There really aren't any direct methods to "paste to relative coordinates" within Illustrator. If you were to show some sample images there may be methods to replicate an offset across multiple objects.
One possible option that does exist would be to use Multiple Artboards, even if temporarily.
- Create the same size rectangle around each of your existing objects, in the same position (align to the object[s]).
- Convert the rectangle to Artboards (Via the Object Menu)
- Select the new object on the first item and
Edit > Cut
(yes cut, trust me)
- Choose
Edit > Paste on All Artboards
The paste will be in the same relative position on each individual artboard. Including where the original object was cut from.
(Which is why Cut
is used and not Copy
- Copy would duplicate that original object in the same position when you paste [you'd get a double-stack where you copied from].)
You can then delete the individual artboards if desired.
If the objects are all identical.....
Offhand.. Symbols
Rather than individual object, if they are all identical, use Symbols.
Symbols are internally "linked" assets. You can repeat a symbol numerous times as opposed to copy/pasting identical individual objects. This makes editing faster and reduces file size because there is only 1 set of objects, regardless of how often a symbol is duplicated.
Create the first object (or set of objects) and the drag them to the Symbols Panel. Then place Symbol Instances - by duplicating the original symbol, or dragging it from the Symbol Panel - for any duplicates you want.
Each instance will be identical to the original Symbol.
To edit a Symbol, simply double-click it in the Symbols Panel. Any edits to the Symbol will be replicated in all instances of that symbol.