Do moving clocks slow down?

Some Russian scholars address this question:

The special theory of relativity has fundamentally changed our views of space and time. The relativity of simultaneity in particular, and the theory of relativity as a whole, still presents significant difficulty for beginners in the theory. v...

The real question is why we can even find real clocks (such as atomic clocks) that work very much like the ideal clocks of relativity. The short answer to this question is that a real clock behaves like an ideal clock because the physics that governs its inner workings is Lorentz invariant. ...

However, this question makes sense in the more ambitious program of Lorentz and Poincar ́e on how the theory of relativity should be developed. Lorentz noted in 1909 that the main difference between the two programs is that “Einstein simply postulates what we have deduced, with some difficulty and not altogether satisfactorily, from the fundamental equations of the electromagnetic field” [109].

Although the Lorentz-Poincar ́e program of deriving the Lorentz symmetry rather than postulating it was never completely abandoned [110, 2], it is clear that Einstein’s program replaced it [109, 111], and for good reason: even today, Lorentzian symmetry cannot be deduced from a more fundamental theory.

The phrase “moving clocks slow down” is universally used and has been in common use for over a hundred years, and we realize that our suggestion not to use it, while correct, is not very realistic due to the ”intellectual inertia” [112] of the physics community. ...

In our opinion, better pedagogical practice would be to base the special relativity from the very beginning on the Minkowski four-dimensional formalism, which allows the introduction of relativistic concepts without any mention of their Newtonian counterparts [13].

It is amusing to see Russian scholars engaging in the same sort of Einstein idol worship that we see in the West, especially when contrary evidence is staring them in the face.

Yes, Einstein simply postulated what Lorentz and Poincare proved. By this, Lorentz meant that they deduced it from experiments, like Michelson-Morley, and established theory, like Maxwell's equations. Einstein ignored all that, took their conclusions, and made them postulates.

Einstein did not dispute this. His original papers did not have any references, but in later interviews he always said  he got the light postulate from Lorentz, and did not use Michelson-Morley himself.

The Minkowski 4-D formalism was invented and published by Poincare in 1905, and popularized by Minkowski in 1908. Einstein eventually accepted it a few years after that.

I am not sure Einstein's program is really so popular in textbooks. All the books I've seen either start with Michelson-Morley, or go straight to 4-D. The books might credit Einstein with these ideas, but he missed them.

For example, here are a couple of recent lectures on spacetime, here and here. Both credit Einstein with discovering relativity, but acknowledge that it was Minkowski's 4-D formalism that caught on among physicists of the day and that still dominates and that Einstein only reluctantly accepted it a couple of years later after arguing against it.

In his 1922 Kyoto lecture “How I Created the Theory of Relativity,” Einstein describes the decisive moment when he became enlightened in conversations with his friend Michel Besso: “My interpretation was really about the concept of time. Namely, time could not be defined absolutely, but is in an inseparable relationship with the signal velocity” [107].

Although the alleged cause of Einstein’s redefinition of time is not known for certain, and it is conceivable that Einstein, consciously or unconsciously, may have borrowed from other authors, most plausibly from Poincar ́e, more than his writings and sayings suggest [108], without any doubt, this new definition of time was the most shocking and paradoxical aspect of special relativity.

Einstein elevated Lorentz’s local time to the status of “true” time, and for his contemporaries it became the successor to Newtonian time. This immediately gave the theory a paradoxical tinge, since Newtonian time is absolute, while Lorentz’s local time varies depending on the inertial frame of reference.

Einstein said his big breakthrough in 1905 was to understand the local time that Lorentz invented in 1895. Lorentz got the Nobel Prize in 1902, and Poincare's nomination credited him with the ingenious invention of local time.

To answer the title question, the clocks do not slow down, relative to their own world lines. They appear to slow down in another frame, because of the 4-D non-Euclidean geometry.

Discussion of whether the special relativity contractions and dilations are real or apparent has a long history. From Wikipedia:

In the period 1909 to 1913 Varićak had correspondence with Albert Einstein[6] concerning rotation and length contraction where Varićak's interpretations differed from those of Einstein. Concerning length contraction Varićak said that in Einstein's interpretation the contraction is only an "apparent" or a "psychological" phenomenon due to the convention of clock measurements whereas in the Lorentz theory it was an objective phenomenon.[7] Einstein published a brief rebuttal, saying that his interpretation of the contraction was closer to Lorentz's.[8]

See also Reality of length contraction. Einstein still did not accept in 1911 that relativity used a non-Euclidean geometry, and that the contractions and dilations were artifacts of that geometry.

Another new paper argues:

There have been three geometrizations in history. The first one is historically due to the Pythagorean school and Plato, the second one comes from Galileo, Kepler, Descartes and Newton, and the third is Einstein's geometrization of nature. The term geometrization of nature means the conception according to which nature (with its different meanings) is massively described by using geometry.

Einstein would not agree with this. He did not believe that General Relativity geometrizes gravity. He persisted in this view, long after he was credited with geometrizing gravity. Steve Weinberg also did not like that view. Strange, as they sometimes used geometric arguments to solve general relativity problems, and almost everyone else accepts the geometry view.