First, this isn't world war one. What you have is your average Motor Rifle Brigade sending about a company near the frontline, say five clicks from the actual 'grey zone', from which they move, using whichever vehicle they can get their hands on (if they can), and move from there. The distance is so that they can try be away from the range of long-range fire assets (even mortars) and right now, mainly drones (also because general marksmanship is on ukraine's side.
From there, they move, they assault something within the gray zone or zones they haven't been before. Most likely? They'll die. Doesn't matter, you can send a few other groups until only five guys remain in the company.
Then, after the Ukrainian platoon manning it is dead or has retreated, the surviving guys can say the took the place, and they get rotated, or die waiting. In any case, the ukrainians aren't going to push to there, so you are going to use that place as a start point to move elsewhere. And you continue like that, field by field.

This is the reason why Russia has so many casualties yet advances.

Why Ukraine doesn't advance? Mobility issue, but referring to aerial coverage, mostly. They cannot move enough assets quickly enough to prevent QRFs in the form of helicopters or ATGM teams to let them demine a section of the front if they wish to assault anywhere, so the places they do attack it's only after deleting the supplies of the Russians. Which of course, makes the Russians spread their ammo across the front to prevent this (most evidently in 2022 v 2024 rate of ammodump hits), and spread units, making them easier to pick and kill, but harder to actually dislodge from a general area. Add that to the fact that any large enough deployment will cause Russia to use its inventory of aerial ordnance to hit your buildup (as you need to saturate a place in order to destroy it thanks to the innacuracy of Russian equipment, and having a general area for them to hit eases their killchain a LOT)