2d or 3d? They vary a bit. Do also be sure what one you are working with -- many games will have 3d backgrounds or use 3d hardware but keep it in a 2d perspective (New Super Mario Brothers being a good example of this). I don't know if any DS games make use of the mode 7 a like options that the GBA/2d video provides.
If doing 2d then you will tend to find things messing with the OAM or the backgrounds (OAM is short for object area memory or sprites, backgrounds tend to be used for text and backgrounds which can be full screen or larger and move around to have some form of background animation, I usually use the space background from tetris worlds on the GBA as my example here). Occasionally someone does something more creative (Ghost Trick: Phantom Detective doing something a bit exotic with hblanks and an overlay) or uses palettes (cycling colours and pulsing light being popular here, though setting something to invisible also works).
http://problemkaputt.de/gbatek.htm#dsvideoobjs http://www.coranac.com/tonc/text/video.htm
There are 2d formats that Nintendo provides, and there is an animation one as part of that called NANR (
https://www.romhacking.net/documents/[469]nds_formats.htm#NANR is a bit old but covers stuff) but it is rarely seen.
The DS was also powerful enough that we started to see more creative cutting of sprites than some older systems where you had some combo of 8x or 16x square/rectangular pixels. Phoenix Wright probably being a good example of the sort of things you might see here. On the flip side the DS also had enough memory and speed that you could plausibly cycle out large full pictures as a crude form of video.
3d.
Here you can animate the model, animate the camera (see something like first person pokemon) or animate the textures, or possibly use fog. Texture animation is limited on the DS compared to many PC games (especially the modern stuff) and is limited more to things like eyes blinking and wheels spinning.
http://problemkaputt.de/gbatek.htm#ds3dvideo for that one.
If the developers are using the NSBMD 3d model format the there is a model animation format called NSBCA that might also be being used. There are also texture animation formats but they are incredibly rare. The animation commands tend not to be too dynamically generated or altered and will usually be close to what the hardware expects. To that end if you can watch it be submitted to the stack you might find where it is.
I cover some of both 2d and 3d in
https://gbatemp.net/threads/gbatemp-rom-hacking-documentation-project-new-2016-edition-out.73394/ (the GBA is similar enough to the DS in most respects that it carries across).