For example, if the Republican party wanted to hold a new primary in 2020 and pick someone other than Trump, is there any specific reason they could or could not do so?


Absolutely, it happens all the time. An incumbent losing to a challenger in their own party's primary election is sometimes called "getting primaried".

Search for "getting primaried" or "primaried out", you can find lots of articles and books:

It has happened to several presidential incumbents as well, according to NPR's When Has A President Been Denied His Party's Nomination?

  • John Tyler, Whig, 1844
  • Millard Fillmore, Whig, 1852
  • Franklin Pierce, Democrat, 1856
  • Andrew Johnson, Democrat, 1868
  • Chester Arthur, Republican, 1884

Of these 5, only Pierce was elected prior to his primary loss. The other 4 took their office in other circumstances (death of the prior president).

  • 3
    Also worth noting that the parties can technically just change their rules and run anyone they want. Would obviously cause a big scandal and very severe repercussions from voters, but technically possible. Mar 7 '17 at 17:49
  • 1
    Right. The "traditional" method, of course, is for another candidate to get more votes within whatever primary method has been already established. But primaries/caucuses are pretty much up to the state party to begin with, so yes, they could just change the rules.
    – BradC
    Mar 7 '17 at 17:51
  • Teddy Roosevelt had a somewhat weird situation as well. The Republicans probably should have primaried Taft in favor of him, but they didn't and it (may have) cost them the WH.
    – Kevin
    Mar 7 '17 at 22:24
  • 8
    Just having someone run against the incumbent usually is a sign that there are issues with the president, in general popularity. Being primaried, whether successful or not, is usually close to a guarantee of change of party in the White House, I believe. Mar 7 '17 at 22:25

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