Whether or not the chest moves, and how far it moves, depends entirely on whether it is hidden in the Border Ethereal or Deep Ethereal.
As Icyfire answers, the DMG explains that every location on the prime overlaps one-to-one with a location in the Border Ethereal, so that we can assume that distances on our prime are equivalent to the corresponding distances on the Border Ethereal (and see Can I still command / control Prime-cast concentration spells while I am in the Border Ethereal? and The Material-Ethereal Isometry).
Thus we can assume that if I put a Glyph on my chest, send it to the Border Ethereal, and immediately recall it without myself moving, the least it will have traveled is no distance, as it has shifted into the Ethereal and back without changing its location on either plane. However, in just recalling it without myself moving I am already taking a risk, since according to the spell the chest "appears in an unoccupied space on the ground within 5 feet of you." Thus, you may have sent it to the Ethereal and brought it back, but before coming back it moved 5 feet while on the Border Ethereal.
If instead I put a Glyph on my chest, send it to the Border Ethereal, walk sixteen feet away, and then recall it, it will have traveled at least 11 feet while on the Border Ethereal and thus will have dissipated the Glyph. I may be able to maintain the Glyph, but only if I am very careful to recall the chest as close as possible to the prime location from which I sent it.
I am assuming in this case that it has gone no further than the Border Ethereal location corresponding to the prime location from which it was sent. However, the spell also tells us that "You hide a chest, and all its contents, on the Ethereal Plane", but does not explain just how it is hidden. Is it, perhaps, simply obscured from view in the "misty, fog-bound dimension"? Then possibly it has not traveled at all and remains in the same spot corresponding to where it left the Prime.
Is it immediately whisked off to some other part of the Border Ethereal corresponding to a particularly desolate part of the Prime such as an empty region of extraterrestrial space? Then it has already traveled more than 10 feet as soon as you have sent it to the Ethereal.
Is it perhaps sequestered on the Deep Ethereal, whose locations do not correspond to the Prime, and where "distance is meaningless" (DMG p. 49)? Then it has moved in some dimension not corresponding to the three axes of the Prime and it is unclear how 'far' it has moved, and it would be your DM's decision as to whether this removes the Glyph or not. Personally, I believe that the spell acts to send the chest to and from the Deep Ethereal, with just a brief transit through the Border Ethereal (where it does not move any distance) before landing on the Prime.
Icyfire cautions, "If it just enters the border ethereal at its current location, then your plan will work, but if it goes somewhere else, then it won't. Given that the text doesn't specify, this will have to be something you work out with your DM." I agree, but would add that if the 'somewhere else' it goes is the Deep Ethereal, your plan might work even better since it might not move in the sense of distance at all.
In a comment to Icyfire's answer, Ilmari Karonen questions whether the chest could simply be hidden at the same location in the Border Ethereal as it left the prime, since when the spell ends the chest can be "irretrievably lost", and if one knew the location it would presumably be easy to retrieve by simply going to that location. Certainly the other alternatives (unknown location in the Border Ethereal or Deep Ethereal) more naturally support the idea that when the spell ends the ability to 'look up' and retrieve the chest ends as well. But suppose a DM wishes to rule that the chest does in fact remain in the same location of the Border Ethereal (whether to permit a stable Glyph or for another reason) - how then might they justify its irretrievable loss at spell's end?
I would point out that one of the effects of the spell is that it hides the chest on the Ethereal. Perhaps one could stand right next to the chest's location in the Border Ethereal, but while the spell is in effect it remains magically hidden. When the spell ends, however, its effects end - rendering the chest now visible and accessible to anything on the Ethereal, native or visitor. If you are quick to make yourself Ethereal when this happens, you might be able to recover it, but any delay will result in the chest having been pilfered before you arrive. In fact, perhaps signs of the chest are apparent in the Ethereal, and it has already attracted a number of treasure-seekers, who can sense that it is in the area, but can't actually find it until the spell ends. This sort of interaction goes far beyond 5e RAW, but it is certainly in the spirit of the original version of the spell in first edition. There, "[N]o creature on the Prime Material Plane can locate the chest even with a gem of seeing, true seeing, etc." Note that the True Seeing spell made "Invisible things and those which are astral or ethereal become quite visible." Thus, even a spell which specifically could see things on the Ethereal could not see a chest hidden on the Ethereal. Further, leaving items in the chest for long periods of time risked them being found by Ethereal beings:
While on the ethereal plane, there is a 1% cumulative chance per week that some creature/being will find the chest. If this occurs there is 10% likelihood that the chest will be ignored, 10% possibility that something will be added to the contents, 30% possibility that the contents will be exchanged for something else, 30% chance that something will be stolen from it, and 20% probability that it will be emptied.
One could assume that in this spirit, a chest left too long unattended is not lost because its location has changed, but rather because it has been found and taken.