When grating, always grate the softest ingredients first, and then the firmer ones. Using this step keeps the grater cleaner.
When it comes to grate for an example mozzarella, the process is not obvious… same apply with butter. The best way to grate these “soft” food items is to place them in the freezer until the get very hard then you can grate the amount you need without the frustrations. Of course when grating your soft food item, you have to be quick.