I could see it playing out somewhat like this:
There could be a legitimate third party that challenges the two major parties, mostly at the expense of the Republicans. The way I see it, the Republicans are broken up into several subgroups:
Religious Isolationists, who believe in limited government so that people can stick with their own. Tend to be pro-School Vouchers and homeschooling, and are much more social conservatives, but don't feel the government should interfere too much. Generally pro-immigration, especially Catholics.
Religious Interventionists, who support things like the Drug War, and are more vocally anti-gay marriage, etc. Generally, both of the above are anti-abortion. These people tend to be more Populists, Mike Huckabee types, and are also more anti-immigration.
Big Business / Big Agriculture - They like regulations written in their favor, subsidies, etc.
Free Marketeers - People who believe in the free market, and have decided that the RP is the best they can do. Pro-immigration.
Right now, the Republican Party is the Hawk party, but that seems to change with whoever is in power. They were against the wars in the Balkins, and had a relatively strong anti-Vietnam segment. I can't see either party keeping the title permanently.
Of course, if the RP were to fall apart, it would affect the Democratic party quite a bit. I would predict that the Dems would shift much more to the Center on Social issues, while absorbing the people I termed 'Religious Interventionists'. I see this as possible only with the decline of social conservatism. I see the Democrats reclaiming the South, with the southern democrats much more conservative, but still populist. The Democrats would move to be more anti-immigration, and would retain their economic interventionism.
During the transition, there would be a period of time when the Democrats were in complete control. I have a feeling that they would attract the lobbyists of Big Business, who would fit perfectly into their system.
This leaves two groups stranded: social liberals, and free marketers. Clearly, my hope is the social liberals and the free marketers band together, and the Libertarian Party wins out, or the Republican Party becomes more like them.