I don't think there is a trick to cleaning the flocking in your glovebox.
It's very delicate stuff, so any kind of cleaning will basically remove it and you'll have bare to nearly bare cardboard instead.
The flocking is basically a bunch of very short and small hairs that are sticking out of some kind of fastener medium like paint.
So they are not very sturdy/robust and if you scrub on the flocking material, it will come off the inside of your glovebox.
Mine was quite old, dirty and crusty looking so I chose to restore/refresh it.
Here is what I used.
Charcoal gray flocking with a med gray craft paint.
It came out great and looks factory.
Click to view attachment Flocking 101.
Have you ever looked at flocking for the glove boxes?
Up close you'll see what looks like a bunch of very short hairs.....what the flock!
It's true; just checked it out under a magnifying glass.
And this flocking material can get everywhere.
Beware! Ha
I used a big cardboard box as a work station.
Once I had sanded the old flock off the inside of the glovebox, I then gave every square centimeter a good coating of the med gray craft paint.
Then into the workstation box went the glove box and and I used the squeeze bottle to liberally coat the entire inside of the glove box while the paint was still wet.
The flocking hairs stick to the paint in every orientation, but a vast majority of them stick out from the paint giving you that nice fuzzy appearance/feel to the inside.
After the paint dries, you just shake out any remaining flocking that didn't get glued up and is loose.
Easy peasy.