More could be found here: http://bitcoin.stackexchange.com/questions/26937/nlocktime-transactions-how-do-they-persist-are-they-broadcast-before-they-are
That answer is confused. Locked transactions cannot be relayed until they're about to become mineable. After they are mineable they hang around just as long as any other transaction, their persistence is not reduced.
The "story" doesn't make a lot of sense, however. If someone is going to give 1000 BTC a year later with no opportunity to take it back, then that is the same as handing over the funds now. All of locktime, cltv, and CSV make sense only if there are cases where the transfer would potentially be changed but is otherwise guaranteed.
Let me make it make sense:
For example, your rich uncle says he will pay you 1000 BTC if you get married and stay married for a year. You don't believe him since he didn't get rich by being generous or trustworthy. So he sends 1000 BTC to a 2 of 3 multsig, with him, you, and the editor of your local paper as signers. Additionally, he writes a transaction spending that payment and paying you directly, locktimed a year from now, and he signs it and gives it to you. You sign it, so the only think keeping it from being valid is the locktime, and stick a few copies in safe places. You go and get married, confident that your uncle will not have an easy time cheating you.
Now, at any point from between now and a year from now you might get divorced and then you can sign a txn giving the funds back, or-- assuming you don't co-operate he can go to the newspaper editor with evidence of your divorce, and get the funds back.
Otherwise, in a year, you can take that locktimed transaction and send it to the network-- with no further cooperation from anyone else. That surprise forklift accident that got your uncle AND the newspaper editor will be no problem for you, their help is no longer required.
In this case the locktime was quite useful-- and there was no reason to send it to the network until the lock was already mature... which is exactly what the network expects.
If the network didn't work this way, people could make locked transactions that wouldn't be good for 100 years, use the network for free forwarding and storage.. then double spend the coins before the lock became mature to avoid paying any fees. Fortunately, there are no interesting protocols that I've encountered where this is an issue.