By the way, you asked, "If an appeals court in another district makes a ruling . . . " Such a ruling would come first from another (trial-level) federal circuit court, not from an appellate court. Of course the appellate courts could get involved later, in any cases where there were grounds for appeal.
A trial in another court, if appealed, would set a precedent in that particular federal appellate circuit. If it reached a conclusion different from the Dover case, it wouldn't change the Dover result.
There are eleven federal appellate circuits, each with its own geographical area, plus the DC circuit. If there's only one appellate case on a particular topic, although its not binding outside of that circuit, it can nevertheless be very persuasive on judges in other circuits.
It sometimes happens that different appellate circuits will come to different results on the same issues (in different cases, of course). In such a situation, the losing side in the most recent case will attempt to get the case heard by the US Supreme Court, to resolve the conflict among the circuits. Many of the mundane cases the US Sup Ct hears are of that variety. The idea is to have uniformity among all the federal circuits.