Rest should self explanatory as it's a pattern.
What apis that I have built and utilize that trigger long running process send back a code of 202 accepted immediately when accepted (validation etc ran).
Then the process is run.
If you need verification that a process is done, you can implement web hooks. Which is just an API that sends data to listening routes. This way you can keep data in sink.
Say your email.
User registers, after all set you send aplicable user data you your API. It verifies the data and replies 202.
Your main app goes on about its business. Then after lumen process the request sends a success webhook with applicable data (if any). Maybe to set a Boolean on the user to true an email sent.
You don't nessesarly have to queue anything on the lumen app.