When your pass expires, the Basic Box consists of the 30 Pokemon that were deposited most recently as of that time (counting ones that are currently waiting to be traded through GTS or Wonder Box, as being more recent than anything else, and when a Pokemon is actually traded by any means, its timer is reset to the moment you received the other end of the trade).
You will not be able to see the summaries of anything you had stored in Home and hadn't pulled out by the time your subscription lapsed; such Pokemon are consolidated into a label at the bottom of the screen that says "Pokemon not in Basic Box: ####", telling you the number but no other details about them. These Pokemon will not be deleted, and still count in the formula that determines how many Home points you receive every day, but you won't be able to do anything with those boxes until you renew your subscription, which you're free to start up again at any time.