Exchange Shell:

Assign Permission – requires logoff/logon before taking effect:
New-ManagementRoleAssignment -Role "Mailbox Import Export" - User "username"

Run Export Request:
New-MailboxExportRequest -Mailbox username -FilePath \\server\share\user.pst

Monitor Status:
Get-MailboxExportRequest | Get-MailboxExportRequestStatistics

Cleanup:
Get-MailboxExportRequest | Remove-MailboxExportRequest

Source: Stevieg