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Is it a conflict to work for a firm defending Bank A and then later work for a plaintiffs firm that often sues Bank A? Is the inverse a conflict as well?

It would be a conflict. Plaintiffs firm could technically screen you, but in the real world if suing bank A was a large source of their income they likely just wouldn't hire you.

OP: reason why I use the bank example is because it is a more consolidated group of major banks. That's rough.