Previous purchases transfer to new console

Transfer how?

If you sign into the same account, all of your games you own digitally will be available to you

If you’re talking about disc based games, they won’t work without a disc drive to validate them

When you purchase content it's tied to your gamertag and the console it was purchased on. When you buy a new console you can transfer the game to the console or redownload it from your download history for free. You can do a license transfer to transfer
the license from the previous console to the new console so that all the games will continue to work the same way (http://xbox.com/drm)

The thing to remember is that if you used a 360 Transfer Cable to transfer your content to another console, only the profile that actually purchased the DLC will be able to use it on the new console eventhough any profile on the previous console was able
to access it before. The way around it is to have the profile that purchased the content redownload it on the new console, eventhough it's all ready been transferred, after this step any profile on the new console will be able to access it again...

I purchased them about 4 years ago on a previous console, transferred them with my transfer cable when I got a new console because the other one died, and then linked the new console. They worked until the account that I had them on got banned.

You need to keep track of your old account, and re-download its purchased content if you get a new console. If you also do a License Transfer, other profiles on your new console will have access to the previous purchases, too (except for Avatar Items).
If you haven't changed consoles, but merely created a new account, you should have access to the old stuff via the console license for the previous purchases.

You can't transfer ownership of purchases between accounts. (If your old account is banned, you forfeited its items when you violated the rules.)