Disclaimer - I only have a YouTube level of understanding of the following subject. 
It is widely accepted the time proceeds proportionally more slowly when gravity is present. For example, when a perfect clock 1km above earth’s surface shows exactly one day of elapsed time, an identical perfect clock on the surface of the earth will show about 10nS less duration for the same day. A practical example of this real phenomena is the need to keep GPS satellite clocks in zero G synced with ground based clocks.
How can we check out this weird situation at home? If we reduce the aforementioned 1km distance to 100 metres and we put an oscillator of 100MHz on top of a 100 metre tower and an identical oscillator on the ground and feed both into our dual trace scope, after one day we should see 36 degrees of phase shift between the two oscillators.
Not everyone has two oscillators with the required degree of stability, least of all me. What then if we have a single oscillator on the ground and a feed coax up the 100 metre tower and a transmitting antenna aiming back down toward us. Then feed to the scope both a signal directly from the oscillator and one received from the elevated antenna. There will be propagation delay of course, but both signals will initially have a certain fixed phase difference.
If in fact time proceeds more slowly at ground level compared to 100 metres above the ground then we should perhaps see the two 100MHz signals gradually creeping out of phase at 36 degrees per day despite coming from the same oscillator source, should we not?
Boy o boy – where do i start!
Firstly, without meaning to offend, u know almost zilch about this stuff, & the little that u do know is wrong. Me myself i know a bit more than u, hence i am happy to discuss this stuff.
1. There is no such thing as time. However, everything that we see & feel is a process, & all processes have a rate of ticking, & that ticking is affected by lots of things.
2. Irrelevant but i will comment anyhow. GPS duz not in any way need the synchronization of any clocks anywhere.
3. U mention a perfect clock on the surface & 1km up. Here u show that u don’t understand (1). If there were such a thing as time, & if Einstein were correct, then u would be ok to use the words that u used. But there is no such thing as time, & Einstein was not correct. Getting to the point, what i mean is that u said "clock", meaning any & all clocks. No. Different clocks will be affected by elevation in different ways. And, having identical clocks duznt get around this problem.
4. U mentioned that problem of stability. That is fair enuff. But i would like to point out that much of the instability of clocks is due to an ignorance of real physics. Much of the instability is due to certain effects, which if understood can be accounted for, if not avoided.
5. Re having a single oscillator & a 100 m antenna. I don’t see how u would get a gradually creeping out of phase (ie that accumulates). What u should expect is a number of different kinds of (nearnuff) cyclical phase changes, due to a number of effects. Some changes will have a daily signature (sidereal day of course), some an annual signature (due to Earth's orbit).
6. Why not use one oscillator. Send a signal from the top of a hill -- & then send a signal from lower. A scope would measure the change in Hz. But i suppose that the heart of the oscillator would be a quartz crystal – a quartz crystal would be affected (by elevation etc) in ways that u could not ever imagine.
7. But i don’t know much about the accuracy of such instruments. And of course the problems of temperature & air pressure etc. But, even if the instruments were very sensitive (& had zero error etc), u could not imagine the (mainly relativistic) effects & peculiar signals that must happen. It would make an MMX look like child's play.
I would be happy to say more about some of this if desired.