How can a mechanical clock tick slower due to time dilation? I was about to watch Interstellar for the third time when I decided to do some research on its phenomena. Time dilation is the one that struck me most. I have read and read about it and understand -- as far as that is possible for me -- how it works. 
But I don't get how a mechanical, or quartz clock could tick slower as they are composed of gears and other parts that are always moving at a constant speed, right. The example with the two-mirror 'light clock' is enlightening but how does time dilation work on a 'real' clock? 
 A: I've rarely seen explicit demonstrations along these lines. It would certainly be very difficult; for most clocks one would at the very least have to use relativistic quantum mechanics.
In general we don't want to re-prove time dilation for every kind of clock. Instead the reasoning runs the other way: if a light clock and another clock are ticking side by side, they'd better also be ticking side by side in another frame, so the other clock must also experience time dilation. Since clocks can rely on classical mechanics, or quantum mechanical effects, or anything else, that means that all our theories should be compatible with special relativity. So we build the theory to be relativistic from the very start. Once you've done that it's not really worth bothering to double-check relativity works in a specific case, as we already know it's going to work for all cases.
Still, this is a neat question, so let's do the check anyway! For simplicity I'll consider a 'magnetic' clock. The idea is that a vertical magnetic field $B_z$ makes a particle move in circles, and the clock ticks every time one circle is completed. In the clock's frame, supposing the particle moves slowly enough to neglect relativity, we have
$$q v B = \frac{dp}{dt} = ma, \quad a = \omega v$$
since we're dealing with circular motion, and combining gives
$$\omega = \frac{qB}{m}.$$
Now consider a frame where the clock is moving along the $z$ direction with time dilation factor $\gamma$. Then the momentum is now $\mathbf{p} = \gamma m \mathbf{v}$, so the only modified equation is
$$\frac{dp}{dt} = \gamma ma$$
and we instead find
$$\omega = \frac{qB}{\gamma m}$$
which is exactly as we'd expect by time dilation. Physically, the reason the clock ticks slower is that the particle is harder to turn around, by virtue of its relativistic motion in the $z$ direction. With a bit of handwaving, this explanation works for the quartz clock too. The gears in the clock have nothing to do with it -- what matters are the resonant oscillations in the piece of quartz, which control everything else. So it's plausible these oscillations get slower as each particle participating in them gets harder to accelerate.
A: It makes sense to talk about time dilation only if you're comparing your clock to another clock that is or has been in different reference frames. The clock is not ticking slower because it is in motion. Otherwise it will be possible to tell if you're moving or not. And the whole idea of relativity is that The laws of physics are the same in all inertial frames of reference. Meaning that regardless if you're at rest or moving with constant velocity in a straight line, you cannot perform any experiment that will determine your state of motion (at rest or moving). So if you're moving with constant velocity in a straight line you will experience everything that a person at rest is experiencing. So the clock it will tick at a "normal rate" in its proper frame. 
A: Say you're driving a car at a constant velocity (constant speed, constant direction) and on the passenger seat next to you is an accurate clock. A stationary roadside observer would measure your time, as recorded by your clock, as slowing down. However, you wouldn't notice your clock doing anything different. Weirdly, you would also measure the roadside observer's clock as slowing down. Hence, the description that moving clocks run slow, one of the fundamental results of special relativity. The effect is barely noticeable at everyday speeds, but if your car was very, very fast (approaching the speed of light) the slowdown becomes significant. The twin paradox (which isn't a paradox at all) explores this notion of moving clocks running slow in greater detail.
