Who said you can kasher meat or milk keilim?
there is a minhag brought down in the mg"a in Hilchos Yom tov not to kasher back and forth from milchig to fleishig, to prevent a mess up. Sephardic are not makpid and even Ashkenazim have several ways around this.
1. Kashering for Pareve use.
2. Treif it up first
3. Kashering for pesach
4. Buying from someone else
In all of these cases (based on different noisei keilim) one can kasher milchig and fleishig keilim.
Weird non-answer.
Anyone know the answer? I assume you have to change the water. How about the pot?
You don't have to switch the water since it's a n"t bar n"t. If one of the keilim is Aino ben yomo you can even kasher them at the same time.
Lechatchila you should not use the kashering water for any other food purposes since you should not be goirem a n"t bar n"t lechatchila (amoung other reasons)