The WebPart's toolbar settings are not preserved during a migration to an on-premises destination or in Normal mode to an Office 365 destination. Settings like the toolbar type of the WebPart will not be preserved causing changes to the Add New link.
After the migration, manually edit the Toolbar Type of the web part at the destination so it matches the value from the source. You can do so by editing the page, then clicking on the small arrow at the top-right of the web part and selecting Edit Web Part.
Use Insane mode if you are migrating to Office 365.