It depends what you classify as "Download Content". I think you might mean stuff like Mystery Gifts and special passes that allow you to get rare Pokemon, e.g. the Liberty Pass for Black & White to get Victini.
If so, nearly all of these events have passed. Most events are time limited, so it doesn't matter if you buy a used game or not, if the game you're playing didn't get the event pass/ mystery gifts, then you can still obtain them if the time frame for the event/ gifts haven't passed. If they have passed, you can no longer obtain them.
Also, if you're lucky, the used game you buy might have some rare Pokemon left in the PC from the previous owner. However, you'd need a second DS to transfer it to another game if you want to keep the Pokemon before restarting the game.