HI iacdxb,
Yes, you can. Please have a look at this file:
https://www.amazon.com/clouddrive/share/P0AZtnxKcLAIbAHq59iheVr7Z3NFacGwHI2QAHGOwtb
Create a MoGraph Selection from one or more objects, then use this selection for a Plain Effector. If there are more in than one in a given selection, they will move the same. Place the Plain Effector after the Random Effector.
For different movements, set up several Selections/Plain Effectors.
Alternative: Just select the one you like in the object manager, or in the editor view, but not that the axis is off based on the random effector. Since it is a Fracture Object they can be moved. However, perhaps make the editable, Fracture active and press c, to just do what ever you like with no off-set in the axis.
All the best