Well you can use the Bad Batch LST on New Batch characters.
Also, I don't think there's any backward compatibility. What I mean by that is, you can't save an old LST in the hope of a character that wasn't allowed at the time becoming allowed due to becoming a legacy character at a later date. (I'm sure I could have written that more concisely!)
"Don't hoard! A LST is tied to a specific list of characters when it's released, and that list will never be updated to include units released later."
Source: https://forums.ea.com/blog/swgoh-game-info-hub-en/game-changes--new-item/12373006
Capital Games Team
