There’s no defined process. The constitution just specifies that the supreme court has appellate jurisdiction except where Congress defines exceptions.

https://en.wikipedia.org/wiki/Jurisdiction_stripping

Since Congress also has authority to actually create and organize the lower courts, they can do almost whatever they please.
The only thing that can’t do is diminish or expand the original jurisdiction of the supreme court.

https://en.wikipedia.org/wiki/Original_jurisdiction_of_the_Supreme_Court_of_the_United_States

So they can’t put a different court in charge of cases involving two states disagreeing over ownership of a river or it’s water, or ambassadors and such.

So if Congress wanted it to be a single use court nominated, appointed and dissolved for one special case they could. Or they could say it just stops at the federal appeals court, the state court or wherever they want.

Personally, I think a single use court established for special high profile cases with a large potential for conflict of interest would be best. There’s some trickiness that would be involved, since Congress can’t actually appoint judges, only the executive can. So if the case involved the current sitting executive (in my opinion only in their personal capacity, as cases involving the office of the president lack the personal liability that makes for a conflict of interest), then they would still need to be the one to make the appointment. Might be able to sidestep it by having the house select already appointed judges without the conflict to hear the case, but it’s very close to appointment with extra steps.

In any case, other than the caveat that’s never happened, it would be so much more clearly unbiased.