I remember noticing that they seemed to be protecting Bush's DNA when he went to a state dinner in some European country (possibly Germany). I wondered about it at the time.
I remember reading a story about when Bill Clinton had a drink in a UK pub, the Secret Service smashed the pint glass he used as soon as he had finished.
Smashing the glass doesn't prevent DNA collection.
Depends how small they grind it, I'd have thought, and whether they take the bits with them or not.