No, a worn chain will NOT slip on a new cassette. Semantics first. The process of a chain sliding over a sprocket without engaging in the teeth of the sprocket is called skate.
Anyway, a chain enters the cassette under slack where it is easy for it to slip onto the sprocket teeth because it can compress or elongate at will to match the tooth profile. As the entry point rotates and increasingly come under tension, you'll find that only one tooth supports chain tension because the chain is too long for the tooth profile and the links behind the one in tension will simply lie slack.
Proof that an old chain does not skate under tension can be found on your bike. As you ride and increase chain length through time, the cassette wears to accommodate the ever-lengthening chain. This only happens in your favourite gears. The sprockets at either end of the favourite zone don't wear. However, when you change to that gear, the chain happily rides it without skating. It is only when you fit a new chain and it skates over those worn gears that you actually discover that the cassette is worn.
It ill be OK for 30 miles or so because even though it will put extra stress on the cassette teeth, it will not damage it on a short ride.
It is all explained in "the book."
