CB Auto Actions acting on the after registration and after profile update triggers with a Code action and Method set to PHP. This will execute your custom PHP and you can update the user object from there using CB API. Example as follows.
Code:
$cbUser = new UserTable();
$cbUser->load( '[user_id]' );
$cbUser->set( 'FIELD_NAME', 'VALUE' );
$cbUser->store();
It can also be done as follows.
Code:
$cbUser = CBuser::getUserDataInstance( '[user_id]' );
$cbUser->set( 'FIELD_NAME', 'VALUE' );
$cbUser->store();
The below might also work with latest CB Auto Actions nightly.
Code:
$user->set( 'FIELD_NAME', 'VALUE' );
$user->store();