Environment: SharePoint Server 2010
Web Application: Claims Authentication - Windows NTLM
User claim token: i:0#.w
User Profile Sync: Configured to retrieve user info from ActiveDirectory. Full and incremental running
Issue: When usernames and email addresses change, I need to manually run Get-SPUser ... | Move-SPUser and Set-SPUser via the SP Management Console to get the user's profile on the web application to change. The Set-SPUser command updates the email and display name of the user. In the User Profile Service Application I can see that the user profile sync has the correct information but it's not flowing down to the web app.
According to: http://www.wictorwilen.se/Post/How-Claims-encoding-works-in-SharePoint-2010.aspx our token is using User Logon Name as the claim identifier. I set the Claim User Identifier (SPS-ClaimId) to sAMAccountName and ran a full profile sync. I watched the MIIS ops update the MOSS profiles with the new attribute yet the changes still did not propagate (even after the user logged onto the site again).
This is using the Windows claim provider. This is not a custom identity provider.
UPDATE - See properties below. AccountName is not updating as expected. 
UPDATE #2 - Ran a full sync and the AccountName attribute on FIM finally updated. However, after the user logs onto the SP site they still see their old information in "My Settings".