One is Silver and the Other's Gold
Compared to the dramatic backdrop of civil war that carried America from the second party system to the third, the transistion from the third to the fourth seems mundane and almost alien (except maybe to some Libertarians); but the effect it had on US politics is no less imporant. It was about coinage; specifically, about moving the US from a currency based around gold as a standard, to one using silver. As with the concerns of slavery in the previous transistion, both major parties were internally split on the issue. But in the 1890s, mid-western silver miners (mostly Republicans), as well farmers who saw inflation as a way out of recession-caused debts (mostly Democrats), finding no outlet for their political aims, began building a coalition, which quickly grew into the Populist party. In 1892, the Populist nominated James Weaver for president... and promply spoiled the election for the Republicans, allowing Grover Cleveland's unprecedented (and unrepeated) non-consecutive presidential terms. (In addition to the 22 electoral votes Weaver won outright, he may have tipped the balance in states worth an additional 69 of the 444 available electoral votes.)
But the story doesn't end there. In 1896, pro-silver Democrats seized control of the party, denounced Cleveland, and successfully nominated William Jennings Bryan, who delived his famous "Cross of Gold" speach; the Populist party jumped on the opportunity and nominated him for their candidate as well. But what was more surprising was that the Silver Republican party, who had split from the Republicans over the silver issue, also supported Bryan. Meanwhile the gold faction of the Democrats, dejected over the result of the convention, nominated their own candidate, although he received practically no votes; instead, many gold Democrats voted for Republican candidate William McKinley. The end result was that many Republicans ended up voting for the Democratic ticket, and many Democrats voted for the Republican ticket, all over currency.
Bryan and his silverites lost, and even the silver issue itself fell by the wayside as economic prosperity returned. But the new coalitions formed in the leadup to the election lingered on, and solidified the supports of both parties. This would lead to decades of almost completely uninterrupted Republican rule.
So we see the same strategy repeated from before: Pick an issue ignored by the major parties, and run on it. The important lesson here is, it doesn't always work, at least not necessarily like you planned. Even though a major party picked up the issue, they lost the election, and the movement died out. Slavery had been an important issue in America since before the Constitution was written; maybe the gold standard just isn't something that captures the hearts and minds of voters, or maybe Bryan only picked up the silver issue to gain votes? Either way, pick your issue, and your allies, carefully.