I followed the instructions given by Synology, but as my actual experience was considerably different (actually easier!) from what supposedly should have happened, I’m documenting them here for reference.
I started by upgrading the old unit to the latest DSM version, and then shutting it down. I moved all four hard drives to the new unit, making sure the same order of the drives in the drive bays was maintained.
I then switched on the new unit and launched the web UI in a browser. This is what I got: