How to change a user’s role

How can I change the role a user currently has without having to do it in the admin dashboard?

I have 2 different roles, each with their own profile form. On occasion a user will want to change roles so I need a way to change it on the front end forms as opposed to in the admin dashboard. I tried adding the Roles dropdown and roles radio fields to the profile form (only visible to the admin role) but it doesn’t change the actual role field. What am I doing wrong?