You can update the Process of an XBert.
If you want these XBerts to appear under a different Process every time, you will need to update the Process against the XBert in XBert Automation.
From the Work board, open an XBert tile by selecting the XBert summary in the Work items list. This will activate a Details Tile on the right-hand side of your screen.
From the Action Bar across the bottom, the Tag icon will allow you to update the Process of an XBert.