Order three more of the shade while they are still being produced. Swap to them when the old ones turn yellow .

Scotch Guard, but still only a matter of a short time and they will be ruined. As mentioned above, but more shades now.

Change out the shades to glass.

Ditto DeWayne.

Ventilation is a health consideration. It isn't just a cleaning consideration. If dirty lamp shades are important enough to you to ask about the issue, then you already know that you should have a proper vent above your cooking surface!

Y'all do know that stoves come with downdraft vents now, My Mom and Dad had one in the 70's and it worked great then, they are much better now.

I disagree. They are fine. They work as well as anything I would have in a kitchen.

That is just a knee jerk reaction from someone who has never had one.

I've had one these last few years and find it inefficient. I have never in my life been able to smell the previous nights dinner when walking into my kitchen in the morning until buying this house. Somehow the one that really lingers isn't seafood like I would have guessed....it's a pot roast with garlic! That downdraft goes under the kitchen to outside with enough force to move plants in the flower bed - some really tempting smells are in my back yard when something yummy is cooking. But the house has radiant heat so no whole house fan and the cooking odors linger.

I had a service person here for another reason last month and talked to him about it. He said no fix, only that the good news is Maytag and Jennair are both introducing a downdraft range that really does finally work well. Available next year.

Hanging fabric over an odor and grease generator is just nasty. So is not venting properly.

To protect them, you could consider applying a high-temperature coating meant for light fixtures. It forms a protective layer that makes cleaning easier and prevents discoloration. Make sure the fixtures are cool before applying anything. Also, regular maintenance helps—clean with a damp cloth or mild cleaner after cooking to prevent buildup. If you're around London, https://tenancycleaning.uk/london/w9-maida-vale/ might have tips or services for maintaining fixtures in kitchen spaces.