Assuming this is a mail server that generates no mail locally:
Once you stop postfix listening on the network it won't get any new mail but will process existing mail it has. With that all gone replace oldbox with newbox, test it and you're done. Inbound email will wait a short time but arrive as expected on newbox.
I wouldn't copy files under /var/spool/ . Make sure any aliases and mailing lists are created on newbox.
When I've done this I've opted to recreate stuff rather than transfer files. That way there's no chance of a config file applying to the wrong software version.
http://www.ibiblio.org/harris/500milemail.html