In the United States the Financial Industry Regulatory Authority (FINRA) is a non-governmental self regulatory organization for regulatory oversight over all member securities firms that do business with the public.
It will help to read Understanding Brokerage Account Transfer Process as the page lists the steps required by both the Delivering and Receiving Firms.
While it may seem everything is automated the Delivering Firm needs to confirm the ownership of the receiving account (name, tax id, etc.) and the Receiving Firm needs to confirm that assets under transfer match what is supported (margin requirements, equity minimums).
Quoting the section on Time Frame:
Once the customer account information is properly matched, and the receiving firm decides to accept the account, the delivering firm will take approximately three days to move the assets to the new firm. This is called the delivery process. In total, the validation process and delivery process generally take about six days to complete.
The Security Exchange Commission page on Transferring your Brokerage Account: Tips on Avoiding Delays lists other types of securities which would cause delay like:
securities sold exclusively by your old firm;
mutual funds or money market funds not available at the new firm;
limited partnerships that are private placements;
Separately I find it interesting that the delivery process is about three days, one day more than T+2 Settlement
2018-06-30 Additional info:
As the transfer process uses ACATS, I was able to find additional pages on the process:
DTCC - Automated Customer Account Transfer Service
Placetrade - What happens once I submit a request to transfer my account (ACAT)?
Placetrade's one in particular does give a day by day timeline.