John Anthony Castro, a 40-year-old Texan, long-shot Republican presidential candidate and the most prolific challenger of Donald J. Trump’s eligibility to be president, has gone to court in at least 27 states trying to remove the former president from the ballot.

On Wednesday, Mr. Castro found himself in a mostly empty courthouse in New Hampshire’s capital, where he was making a second attempt to advance his arguments; his initial case was dismissed last fall.

None of Mr. Castro’s lawsuits have succeeded. But the New Hampshire case is part of a growing constellation of ballot challenges — some lodged by established groups with national reach, many others far more homemade — that have been playing out in more than 30 states. Challengers in Colorado and Maine have succeeded, at least temporarily, in getting Mr. Trump disqualified, while other lawsuits have stalled or been dismissed. In at least 22 states, cases have yet to be resolved.

All the litigation has made for an odd, diffuse process in which some of the weightiest issues of American democracy are being raised not primarily by elected officials or a political party, but by an unlikely assortment of obscure figures, everyday citizens and nonprofit groups. Even some of the players are wondering what they are doing there.

“How did we get to this point, where you have random brewers in Wisconsin throwing Hail Marys to try to get Trump off the ballot?” said Kirk Bangstad, a brewing company owner and liberal activist who filed an unsuccessful challenge to Mr. Trump’s eligibility with the Wisconsin Elections Commission. Mr. Bangstad, who is now considering a lawsuit, readily admits that he wishes someone more prominent would have taken up the cause.