Why do most tenure at an institution less prestigious than the one where they began teaching, and than where they received their Ph.D?

This is called the "law of descending prestige."

• Most universities hire the most prestigious faculty they can.
• The number of PhDs from the top universities is larger than the number of job openings at universities.
• Therefore, so long as PhDs seek employment at universities, most of them will get jobs at less prestigious universities than their PhD university.

It is not "inexorable," just probable.

Let's pick a real world example. MIT produces around 500 PhD graduates a year. At the same time MIT hires around 50 new professors per year. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4309283/

That means at most 50 MIT PhDs can find a job at MIT. 450 MIT PhDs will have to find a job somewhere else... Which answers your question! Obviously, MIT is not the only prestigious school in the world (or Cambridge, MA for that matter...), but this argument is independent of how many schools are included in the prestigious circle. It all boils down to the fact that (on average) each professor will graduate way more than one PhD student over their lifetime.

This example illustrates what happens in an academic job market in a steady state as is approximately the case at the moment.

Regression toward the mean.

Basically, if you're already at the top, the only place you can go is down. It is well known that over many trials, individuals who perform very well in early trials will tend to perform worse in later ones. This doesn't suggest that the individuals themselves are getting much worse, but instead reflects some aspects of the random nature of success. An illustrative example can be seen in baseball, where players with the highest batting averages in one season tend to have lower batting averages the following season. That doesn't mean that all the top-performing players have all gotten worse; instead it's a reflection of the fact that almost all individuals have worse batting averages than the best averages from any one year. There's some element of random chance in being "the best", and it's simply unlikely to consistently come out on top.

We can view this scenario as a series of trials, where individuals earn their degree, get a professorship, and then become tenured. It's simply unlikely that someone who succeeds at the first step will succeed at all three. Even if you've earned a degree from the top 1% of academic programs, you still have a very low chance of getting a professorship at the top 1% of schools. Unless all graduates of a top 1% school earn a tenured position at a top 1% school, it must be the case that top 1% graduates, on average, get tenured at worse universities. Conversely, if you earn a degree from the bottom 1% of academic programs, practically any professorship you earn will be at a better school.

The overall scarcity of tenured professorships exacerbates this problem, but the phenomenon would likely still exist even if there was a tenured position for every PhD graduate. By chance alone, graduates of "the best" PhD program will become tenured elsewhere that's worse, while graduates of "the worst" PhD program will become tenured somewhere that's better (or not at all).