I was having trouble finding this, so I thought I would ask; does moving items on Portfolio for JIRA change the Rank value in the same way the Scrum board does?
Yes, and conversely, changes on the team's board will also change the rank of tickets displayed on the Portfolio plan. However note this only applies to Epics and Standard issues (Story, Task, Bug) - for virtual hierarchy above Epic, the Rank rules inherit from the Epic and Standard issue types. Ranking something in the virtual hierarchy above Epic does not always properly update the rank of Epic and Story below it globally.
I wish there was a way to have rank fixed for a certain level in the hierarchy, but this would be very difficult to solve for.