In an effort to defeat Grant, the Democratic Party accepted the nascent Liberal Republican Party’s platform of and candidates in the 1872 elections. In OTL, they still lost. But, if the coalition had won, likely due to someone other than Greeley being the nominee, what would happen to the Liberal Republican and Democratic Parties? Do they continue to fuse together in opposition to the Radical Republican faction of the earlier Republican Party? If so, do we see the Democrats being subsumed by the Liberal Republicans?
I’d imagine if this were the case we’d see them just become the Liberal Party rather than having two Republican Parties.
With the Liberal Party being dominated by classical liberalism, does the remaining Republican Party become more populist and progressive as a result?
No, because the progressive movement in the late 1800s was about anti-corruption and anti-bossism (this is not the same progressive movement of the 20-21st century), and the remaining Republican Party (the Radical Republicans, who later became the Stalwarts and Conklinites) were much more tolerant of patronage and bossism.
The urban machines and the Reconstruction governments, which were both legendary for their corruption and tolerance of patronage politics, were going to cleave hard to the Radicals and Stalwarts irregardless. If the Liberal Republicans live on, their focus on corruption would split the Republican Party.
So the remaining party does not become more progressive, as the progressives would ally themselves with the Liberal Republicans, or more likely, form their own faction.
As for populism, I don't think you're going to see much of that until the silver issue becomes paramount. Or, rather, you'd see it form a lot quicker in the rump Democratic Party. The Bourbon Democrats in this case would be mostly Liberal Republicans, and they would quickly get outflanked on the race issue by the rump Democrats, who would be populist in tone. This is what happened in the 1890s, where Pitchfork Ben and the Redshirts and others overthrew the Bourbon Democrats, partially over economic issues (the populists obviously wanted more egalitarian, silverite economics), and partially over lynching (which became huge in the 1890s especially, and the populists wanted less anti-lynching legislation).