Automating the Shopify Dropshipping process has become easier as the years have passed. There are several avenues for automating, the first within Shopify itself and the second by third party apps.
Automating within Shopify
Email Drips: Within Shopify you can create an email drip for free. You do this by going into Settings and then into Notifications. In the Notifications section there are around 20 methods by which you can create a series of emails to send out to leads and customers or to anyone who has opted into your website or abandoned carts. You would send these emails out in a "drip"like fashion. Meaning the first email you will send out after 10 days of the person interacting with your website and providing you with an email address, then you will send the next email 2 days after that and so on. There is a psychological science to the intervals in which you send out drip emails. Sending out emails also depends on the reason you are sending the email to your lead or customer in the first place. For example, if you are sending an email series due to an abandoned cart you may feel that you can send an email up to 9 times in a month to remind your potential customer that you exist. If you are sending an order confirmation you could send a couple saying their order is confirmed and then a couple when their order is being processed and packaged and then a couple when the order is in transit and then a couple when the order is expected to be delivered.
Third Party Apps: Since Shopify Dropshpping started, third party individuals and companies have created apps to be compatible with Shopify to automate internal processes even further. Oberlo, Theme Deploy, Sellbrite, Intuitive Shipping just to name a few. These apps are available some free some for a cost to automate, simplify, beautify and inform on the different aspects and components of your Shopify Dropshipping store. Shopify's website has extensive lists of apps to browse and choose from.