Select the part / circuitry you want to copy
Edit ► Copy
Select a Reference Point, it is typically recommended to use Origin.
Open the New PCB
Edit ► Paste ► Paste Special
Enable Keep Net Name ► Paste
Select a Reference Point for the Paste.
For the Schematic, you can simply select the primitives:
Edit ► Copy.
Before pasting into the new schematic document, we will disable the option to Reset Designators on Paste.
Disable the option:
Preferences ► Schematic ► Graphical Editing ► Options ► Reset Part Designators on Paste.
Once you have the two documents with the copied primitives in the same project, you can link them by: Open the PCB ► Project ► Component Links.
(Note: Component Links are a link between a component in the schematic and the corresponding footprint in the PCB. Cutting and pasting a schematic component will break this link, requiring re-linking as shown below. This step is necessary before any designator changes are made. To move components from one page to another in the same project without breaking the link, use Edit ► Refactor).
Then select the Add Pairs Matched by: Designators in the bottom left. Perform Update.
Once the links have been updated, click OK and run the ECO from Schematic: Design ► Update PCB.