If you are manually updating a user within OptimizeMember and want that user to be subscribed to the new list that is associated with the level you are adding them to, you need to ensure that you tick the option inside the User modification screen that looks like this:
"Re-process List Servers: Yes, send a mailing list confirmation email to this User."
On the user modification screen - (found by going to OptimizeMember > Browse Members - select the member you want and click edit) - you should ensure you tick this box after you have changed the user role/level.
This will subscribe the user to the list/campaign associated with the level you have selected on the user modification page. You will need to click the "Update User" button for this to trigger the notification, and your user will be required to confirm their subscription if your list uses double-optin.
Please note that we cannot automatically move a user from one list or campaign to another within your mailing list service, this process works as if the user is a new subscriber, so it will depend on the service you are using to how this is treated.
How to configure Email Lists for Levels inside OptimizeMember
If you're not sure how to configure a list inside OptimizeMember for a certain user level, you can do this by going to OptimizeMember > General Options > Define Mailing List/Autoresponder Configurations
You will need to have configured at least one autoresponder integration inside the OptimizePress general settings and ensure you have OptimizePress active otherwise you will break this feature.