I think largely it would involve gaining a lock on the demographics already trending Democrat, that is suburban white voters, and enormous margins with younger voters who will become a larger part of the electorate as they grow older. Geographically I think a key election to look at would be Alabama Senate 2017. Counties like Tuscaloosa and Madison would have to be consistently (non-Atlas) blue, and minority turnout would have to be consistently high. For Democrats to actually dominate a party system like this, results like in Alabama would need to be a consistent thing in other now reliably red states. Counties like Tulsa OK and Greenville SC would have to flip, with Republicans really having any standing only in white rural areas.
Of course this is all hypothetical. The counterpoint for Republicans would be to look at Charlie Baker and Larry Hogan win maps in Massachusetts and Maryland. Essentially it boils down to if either party can completely win over white suburban voters, then that party would dominate.
If Democrats can make that sort of coalition in the south sustainable combined with the religious right fading in relevance, say by succeeding in getting roe v wade overturned and essentially winning on abortion after they already lost on gay marriage I could see it.
The religious right is organized around very broad based racial and cultural grievances, not the singular issue of abortion, which conservative Protestants discovered they cared about in the 1970's. So that would never fly. Moreover, Northern cosmopolitans would never accept such an outcome, so the fight would continue indefinitely.
Two things I heard-
1) It was either Jerry Falwell or Pat Robertson who once said that abortion was a Catholic issue as late as 1970?
2) Any outcome where Roe gets overruled PROBABLY makes abortion a bigger issue at least for a few years. Until and unless secular progressives or traditional nationalists are willing to say "it's the law of the land. It's that way for a reason, OK? Period!" and cede the issue , we will see an major policy changes causing a lot of resistance.
3) It will be a HUGE task to get working class rural people to assimilate more into broader society, whether they are minorities or white. Especially to the point where they just see themselves as working people at the level they don't feel guilt voting for their self-interests.