Your weapon upgrades, status upgrades (those that you get by spending Argent cores) and armor upgrades are not carried over to the Rune Challenges.
The Rune Challenges always throw you at situations where you are at a specific state defined by that Rune Challenge.
Most Rune Challenges don't even give you a gun with upgrades.
I remember at least one Rune Challenge that gives you the Combat Shotgun with the Charged Burst upgrade, and I hadn't this upgrade at the main game at all.
There's also a Rune Challenge where you start with 1 life point and killing enemies using Glory Kills doesn't give you health pickups, so sometimes, the Rune Challenges don't follow the main game mechanics for example.
They are basically a completely different game mode within the main game.