When items go to the border ethereal, they don't have to move.
DMG 48 says that every location on the material plane overlaps with the corresponding location on the border ethereal plane:
The Ethereal Plane is a misty, fog-bound dimension. Its "shores," called the Border Ethereal, overlap the Material Plane and the Inner Planes, so that every location on those planes has a corresponding location on the Ethereal Plane.
Therefore, if you're simply moving the chest from the material plane to its corresponding location in the border ethereal, there is no movement per se.
This interpretation is consistent with the Ghost's etherealness ability, which allows it to move freely between the two planes (MM 147):
The ghost enters the Ethereal Plane from the Material Plane, or vice versa. It is visible on the Material Plane while it is in the Border Ethereal, and vice versa, yet it can't affect or be affected by anything on the other plane.
Note that while doing so costs the ghost an action, it doesn't cost it any movement. Additionally, while it's technically on another plane of existence, the Glyph of Warding spell doesn't address different planes, so I suppose it doesn't apply.
Leomund's Tiny Chest is unclear
It's clear that if you summon your chest to yourself while more than 10 feet away from its original location, your Glpyph of Warding spell will end. However, the text of the spell is a bit ambiguous as to where exactly on the ethereal plane the chest goes when the spell is initially cast.
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.