When a user is not able to update any permission sets on the Mass Assign Permission Sets tab, you need to verify that the logged-in user's profile has the permission named "Assign Permission Sets" checked, under the administrative permissions section of profile.
- Go to a specific user's profile (for example, Custom: Support Profile). Click on the Edit button at the top.
- Scroll down to the Administrative Permissions section and Check the checkbox of Assign Permission Sets to assign the permission.
- Scroll up to the page and click the Save button.
Note - This permission can also be assigned using permission set instead of a profile update.