This can be done but it involves quite a bit of design work.
We have a "Clone Incident" action that will prompt for a user name then create a child incident with all the same information as the parent (including status & assignment).
To get this to work you need to create a new object "Clone" and make it a collection on incident. It has to have the attributes on it that will be different from the parent - in our case just raise user.
Have the process run the clone action (to allow you to enter the raise user) and then after it have an automatic action of create child. On the create child window copy all the data across from the incident except for the raise user which you copy from the clone collection.
Don't bother having the clone collection showing on the incident window as it just confuses the issue.