When you enable Exchange Online Archiving for a user on Office 365, it can take 24 hours or more for the Managed Folder Assistant to start it’s first run and begin archiving email out of the primary mailbox.
If you want this process to start immediately, connect to Office 365 with PowerShell and tell the Managed Folder Assistant to start running.
$UserCredential = Get-Credential $Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://outlook.office365.com/powershell-liveid/ -Credential $UserCredential -Authentication Basic -AllowRedirection Import-PSSession $Session Start-ManagedFolderAssistant -Identity firstname.lastname@example.org
Then wait 5 minutes or so for it to begin running and check the progress via logging in to OWA and looking at either the contents of the In-Place Archive mailbox in the left-hand sidebar or via the Exchange Admin Centre > Mailboxes > Recipients > In-Place Archive > View details.
Don’t forget to disconnect the remote PowerShell session by running the following command.