I stumbled across this question when planning for a Claims-mode enterprise portal site: how you would go and change the password for a claims user?
I looked at the template page generated in the logon site and noticed it has the normal mechanism of a password reset, however, what I was interested was a way for an administrator to change the password. Plus, I must admin I didn't spend much time researching this, but I wasn't looking for a way to allow the user to reset it's password based on a question&answer combination.
So, going back to how to achieve this, in the aspnetdb you have all these stored procedures that are actually allowing you to manage the records in the tables.
I played a little bit with them and, for instance, the AX's New-AXUser PS cmdlet is calling the aspnet_Membership_CreateUser sp.
From here you can simply notice the aspnet_Membership_ResetPassword and aspnet_Membership_SetPassword. Use either one of them to work with the password management. I was thinking to write a small wrapper in AX to call them and manage the claims users' password from within AX.