How to store the position of various objects after the exchange of layout, and return to the current layout?
Set the object to global:Yes.
Be aware though, that if you swap between two or more other layouts, it will move those objects to other layers (within the base layout) occasionally.
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You can use load and save actions
It does not work well with overall placing: yes.
I would not have another option to solve my problem?
The objects are moved, hence when returning from a layout to another object should have the same position that was left.