Don't wash it, don't use soap or a water based cleaner.
Buy a large bottle of White Spirit from a DIY shed or paint shop.
Roll some clean cotton rags into a ball and pad the ball with kitchen roll.
Go outside.
Shake the feathers away from the stain.
Put the padded ball on the other side of the stain
Put on plastic (not rubber) gloves.
Make a pad of kitchen roll
Pour white spirit into a saucer or small dish
Dip the kitchen roll in white spirit and dab it on the stain.
It will start to dissolve the grease and some of the dirt will soak into the ball behind.
Turn your pad and use a clean surface and dab it again (do not dab the stain with a dirty patch)
Do not rub.
When the kitchen roll on the ball underneath is dirty, take it off and put on some fresh.
When there is no more dirt coming through onto the pad, and there is no more grease, you can stop. Let it dry outside. There will still be a faint mark of ingrained dirt. Hang it in the sun to dry.
You can then brush a detergent stain remover into the mark and and sponge it. I presume you will not want to launder it yet if it is feathers.
White spirit is flammable and attacks rubber. This is why you must work outside and let it dry completely before putting it in the washing machine. If it has not fully dried it will make your washer smell.
leave your pad outdoors to dry. When it is fully dry, wet it with water and detergent and seal it in a plastic bag and dispose of it. The water will prevent it catching fire.
Or, instead, when it looks dry, you can use it as a firelighter to start a barbecue. Do not put liquid spirit on or near a flame.
You can also use kitchen roll, plastic gloves and white spirit to clean excess grease off the stairlift. I believe installers charge you to remove them, then sell them on to new customers.
You used to be able to buy dry cleaning fluid in a small bottle, branded Thawpit, but I don't think it is available now.