When you archive a message, Inky sends an "archive" command to your mail server which sends the message to that email provider's archive folder. In turn, Inky mirrors the archive folder (basically syncs the email provider archive folder on the email server with what's on your Inky device). If you were to go into a gmail account for example the archived mail should be locally in "All Mail".
Inky comes with pre-built mailboxes in the home screen, at the top of the left pane you will see additional filters. There's a mailbox called "Unified Archive" where Inky will display all of the archived mail from all of the email providers you've added to Inky. "Unified Archive" is not displayed by default in the "Mailboxes" menu with "Unified Inbox", "Filtered Inbox", etc... Here is how to get to it:
In the "Mailboxes" menu bar click "Edit", then "ADD MORE MAILBOXES". There are a number of additional pre-built mailboxes you can add to the homescreen if you wish, and click "Unified Archive".
Alternatively if you just wish to search against that mailbox:
To get to "Unified Archive" mailbox- you can select any of the pre-built mailboxes on the home screen (For example: choose "Unified Inbox" mailbox). At the top of any mailbox you will see additional filter options. Click the folder icon (highlighted) and scroll down to and click "Unified Archive" to display the contents of all your archived mail across all your accounts.