You may already have noticed, seeing the previews or messing with the models themselves, that there are a few more complicated features that are made to workaround SFM's limitations and push it to its limits.
All of these involve animated textures and extra textures that would not fit in a singular, few-dozen list of skins. I will address them here.

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Models such as attack_move.mdl and enemy_move.mdl have from several to dozens different text names for different moves. Don't worry. I textured them all.
Make sure you've subscribed to this part of the pack before proceeding:
https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2836110965
I had to put it into a new pack because SFM won't pack textures that aren't already connected to a model's .vmt.

These custom textures will be located under the model's original materials file, within a new folder called "names".

1. Right-click your model in Animation Set Editor. Select "Override Materials". Right-click again, and select "Show in Element Viewer" > "Model".
   
2. Under the green name of the material, create a new attribute: a string named "$basetexture".
   
3. Fill in the $basetexture with a file path to the texture. The path starts with this:
   
   models\samlovesmath\fwmoves\
   ... and ends with either one of these:
   
   attack_move\names\ ...
   or
   enemy_move\names\ ...
   ... and the file name. It usually looks like this:
   
   move_mictoss.vtf

attack_move uses a weird thing to make it's shadowed, transparent font, so you gotta put it under move_name AND move_nameshadow, too.

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When SFM has multiple objects with a similar animated texture, all objects will always play the same frame. This may not look good and may limit creativity.
I want to animate frames. Do you, too? Good. I found a solution after searching on Google for a while.

Credits to Detonatress here on Steam for finding a solution, and using it on his TV model. I tweaked it just a bit but now you can easily edit the textures just by using Element Viewer.

1. Right-click your model in Animation Set Editor. Select "Override Materials". Right-click again, and select "Show in Element Viewer" > "Model".
   
2. Under the material dropdown, find the texture you wish to edit/animate. You should make 2 new attributes by right clicking the green material:
   ----a. a bool named "$manual", and
   ----b. a float named "$frame".
   
3. Checking (enabling) manual will override the automatically-playing texture, and allow you to edit the frame by changing its number.
   
4. If you wish to edit this as a value in the Graph Editor, right-click the green material again, and select "Create AnimationSet for Element".
   
5. You will be prompted to enter a minimum and maximum framerate, to set slider borders. You can find this number in the material's .vmt, or at the end of this section in the table.
   
6. Ensure "Show Scene Hiearchy" in the Animation Set Editor is disabled. You won't be able to find the newly-created slider unless you do.

Here is the table for framerate values you should enter: