There are some pitfalls you should be aware of when using this function.
The script will still occupy a FPM process after fastcgi_finish_request(). So using it excessively for long running tasks may occupy all your FPM threads up to pm.max_children. This will lead to gateway errors on the webserver.
Another important thing is session handling. Sessions are locked as long as they're active (see the documentation for session_write_close()). This means subsequent requests will block until the session is closed.
You should therefore call session_write_close() as soon as possible (even before fastcgi_finish_request()) to allow subsequent requests and a good user experience.
This also applies for all other locking techniques as flock or database locks for example. As long as a lock is active subsequent requests might bock.