Natural fibre garment lining

I've been thinking about the fibre content of my clothing and really consciously avoiding synthetic fibres for anything I'm adding to my wardrobe. What do/did people do about lining garments when synthetic polyesters and the like aren't an option? Is real silk the only way to go here? I'm thinking especially sleeves. I already don't like wearing long sleeves, but trying to jam a long sleeve into a jacket sleeve that doesn't have a slippery lining makes me cringe.

Looking at Victorian and earlier garments (mostly the UK, Western Europe, and Canada since that's my area of historical interest), there are a few main methods.

1. is layers.  Linen close to the skin, wool and cotton the further out.  

2. flat lining - so the two pieces are treated as one.  I did this with a skirt and some cotton sheets and it's actually very smooth.  Nothing like the friction and static that I get from bag-lined clothing.  I could easily see having a sleeve lined that way.  

3. Also, linen can be very smooth - they sometimes polished the fabric with what looks like a warm paperweight (stone or glass) to make it slippery.

4. for higher end garments, silk was often used on the sleeves, like how my current commercial coat has cotton lining for the body and synthetic for the sleeves.

Good quality cotton lawn is an option. It tends to be silky, is commonly used for lining garments, and can be used in areas that need extra "slip" like sleeves. I have some Kaufman Sophia Washed Lawn in my stash for this exact purpose - lining the sleeves of a jacket. The low friction areas of that particular jacket (fronts and back) will be lined with flannel for warmth.