Typically complaints about violation of ethical rules of conduct and attorney discipline are made by clients and investigated by the State Bar or Court administrators and proceed to hearings if the investigations determine that there is reasonable cause to believe rules have been violated.
In the situation you are referring to, which isn't entirely clear, it sounds like a "law firm" is "repeatedly sued for (somehow) violating state and constitutional rights (presumably on behalf of a client, law firms not being themselves plaintiffs or petitioners, perhaps, unless they are suing a client for fees).
Perhaps you are saying that this law firm brings "frivolous" actions on behalf of clients, actions without merit to harass and annoy, repeatedly and loses, like SLAPP suits (e.g., lawsuits intended to silence critics exercising their free speech rights in public forums). In that instance, the Courts usually impose sanctions or penalties, including orders not to file further suits of that nature or award damages or attorneys fees to the prevailing parties.
However, the situation posited in your question that there is something "automatic" in this situation that the State Bar will consider this an ethical violation or will investigate this on its own motion without a complaint from a client or court is unlikely. That's usually not the remedy; the Courts usually do one of the other things like sanctions outlined above rather than complain to the state bar.
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Because the brief facts set forth in your post are confusing, I can't help but wonder if you are working with info from some sort of index in which the law firm is repeatedly named because it is counsel in a great many cases.
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Are you maybe confusing the difference between a firm that is repeatedly sued with a firm that repeatedly represents clients in constitutional cases?
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