In summary, they all start with pulverized stuff from which lithium carbonate is extracted and turned into lithium hydroxide. The fact that it's batteries versus rock just changes the front end; nothing downstream could care less.
[1] https://catalysts.basf.com/blog/lets-talk-recycling-what-is-...
[2] https://samcotech.com/what-is-lithium-extraction-and-how-doe...
[3] https://en.wikipedia.org/wiki/Lithium_hydroxide
[4] https://www.cmegroup.com/markets/metals/battery-metals/lithi...