More detailed shipping information is available in the field called fulfillments.events in EZ Exporter.

To export all the delivery dates (if an order has multiple fulfillments), you can select the field fulfillments.events.delivery_dates to include them in the export.  The dates will be separated by a comma if there are multiple delivery dates.

To export just the final delivery date, you can select the field called fulfillments.events.last_delivered_event.happened_at.  You can also include the message associated with the delivery event by selecting the field fulfillments.events.last_delivered_event.message.

One thing to note is this information may not be avallable for certain shipping carriers (e.g. DHL).