You install it like anything else and just launch the game normally.
Remember that DLC is something you download and it detects and uses, whether it's an eShop copy or a cartridge one.
The only issue would arise if you tried to use CIA DLC with Gateway Classic Mode, which then would not work.
devmen, BigBlueMenu (which is just a newer devmen), BigRedMenu (clone of BBM which isn't Nintendo code), FBI (another open source one)... whatever.
DLC isn't a special case like installing older system titles where the older devmen is the only one that "just works"; anything can install it so long as you're in GW/CFW to actually enable installation.
DLC is not tied to a specific version of the game, it installs to the 3DS. As long as you're playing that specific game, the DLC will work for it. Can be .3ds or .cia
Just know that DLC is region specific, you'd need to use Mario DLC (USA) with Mario (USA) for example.
Both .3DS and CIA versions of games use the same Title ID, and so the DLC applies to anything using the proper Title ID. Different regions will have different Title IDs, so if you have the US/NA version of a game, you need the US/NA version of the DLC. Once the DLC CIA is installed, just run the game normally, and it should find and use the DLC.
Certain actions can cause DLC to be removed, such as visiting the eShop. But the DLC can just be reinstalled afterwards.
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