Hi,
it turns out my answer wasn't entirely correct, here is more information:
The field needs to be assigned to the "Search Form" of the Search Type. "Export" button is a "button_submit" field type. From the field parameters you can select "Exporter" as task and then choose one existing session (the session must be created from the Exporter back-end first...)
And about security - as written there is permisison on the field itself, but you also need to set the permissions of the Content Type (depending of the Content Items they have on the List) - there is an "Export" permission there