I think the Republican party will go the way of the Whigs if they continue on the course they're on.
While it was an ongoing movement since at least Reagan, Trump really pulled the party so far to the authoritarian right that more and more establishment republicans are revolted about this movement and revolting against their party to some degree, just see The Lincoln Project or Republicans Against Trump for instance. If the core continues that way, I could see a wave of republicans changing their party affiliation to either democratic or libertarian and the GOP becoming a minor party as a result.
This would also pull the democratic party somewhat to the right, leaving amply space for a new party to the left of the democrats.
I think it is far more likely that the Democratic Party dies. The Whigs died because they were too large of a tent that couldn't take a decisive stance on slavery. Republicans, being a much smaller tent party, are able to hold more ideological and party cohesion than Democrats. Establishment Republicans really don't have the popular support base to "revolt", to be honest. They lost their credibility with the rank and file Republican during the Bush Jr. presidency.
Since Trump took office, 18 Republicans in various political offices have changed their party affiliation to Democrat, 3 became Libertarian, one part or the conservative party, and 10 became independent. That's 32 persons in less than 4 years.
At the same time, 12 Democrats became Republicans, 3 joined the various Green parties (each state has it's own), one became Libertarian, and 5 went independent.
The thing is, since the 1960's there were far more democrats that became Republicans than the other way around, but the trend is shifting now. That's why I think it's the republican party to go away, but you could be right.
The difference between an establishment Democrat and an establishment Republican is pretty thin. I could very well see both merging together, either as Republican or Democratic party, and see both leftists and Trumpism leave on the left and right respectively.
Fun fact I found out while researching the numbers: Donald Trump changed constantly his party affiliation. He was also a Democrat, Independent, and Reform Party member at various times during his life.