Feature Request: Partial Event Registrations (like abandoned carts)
It depends on the scale of the partial registrations you have, but I don't think you need any additional functionality. We get a small number and I think they are best followed up manually. All of them have been people failing to complete the transaction in PayPal. One was them was someone who fell asleep (!) and the rest were people uncomfortable with anything online and not understanding PayPal and a friendly personal email was the right thing where an automated message would probably have not helped.

You can Find Participants with status "Pending (incomplete transaction)". Also I think you need to do it manually so that you can see if they tried again soon afterwards and were successful, which has often been the case.