What I can't see is any (legitimate) way to tell them that you want somebody else's direct mail to stop coming to your address, as they no longer live there.

The printed form with which I registered did have that as an option; you could try requesting a pack.

I lived in a flat in Oxford where a free nappy would arrive periodically for the previous occupant (or her child). We never did think of a suitably droll use for them.