In general, the premise that a fringe can push people out is a bit flawed... it's more that you have a platform that only the "fringe" or "radical" arm still identifies with, while it no longer appeals to people who are "moderate." They're not forced out by the fringe, per se... it's that the fringe still finds the home habitable, while the other end of the party thinks the one down the street better reflects what they want. But for argument's sake let's just accept that yes, moderate Republicans are identifying more and more with Democrats, Libertarians, and independents every day. I don't even know if that's debatable.
So now let's talk about if the concept of people being pushed out long-term is a real thing, and if there's actually a chance for the "downfall" or "collapse" of a party. According to my American Congress professor at ND, there is no scenario where a mass exodus from a party to another in our two-party system is more than temporary. In our two party system, a fringe cannot logically dominate the identity and platform of a party for an extended period of time. It's not sustainable on many levels.
The moderate Republicans may get "pushed out" in the short term to "independent" or "democrat"... but then what happens is as you push more and more towards one party being unpopular and the other being overly-popular, you get implicit redefinition of the party platform such that the population is once again close to "evenly" split.
Imagine that lunatics in the Republican party continue to force out moderates to the point where 67% of the country identifies as "democrat." At this point, you've effectively got people segmented 1/3 Republicans, 1/3 Democrat-moderate, and 1/3 Democrat-liberal. The "Republican" third would fail to have any clout, and simultaneously the Democrat blob would not be homogeneous enough. What happens every time in the history of two-party representative republican (not the party, the system) government is that as you approach this point the fringe loses their grip/influence, and the dwindling party gradually and seamlessly adopts a platform that is more popular. This is why in our two-party system there has also never been one perennially dominant party and the dominance is cyclical.
So let's talk for a second about a concrete example of an issue that is driving people away from the Republican party: gay rights. Once it gets to a point where a vast majority of people believe in certain gay rights, then it is no longer an issue where each party will pick a side. There will not be a "pro rights" party and a "anti rights party"... both parties will be "pro rights" in their platform. And once that issue comes off the table, a hypothetical gay people or pro-gay rights people driven away from Republicans because of that issue who agrees with whatever else is in the platform will gradually drift back to the party. And so on and so forth with every issue until all of the beliefs held by the "fringe" aren't part of the party's identity anymore.