A fast and easy fix would be a bar of hand soap rubbed on all the the moving parts then clean up with a clean dry rag or paper towel.
A better fix would be to make up a batch of your own furniture polish.
All you need is a bottle of mineral oil and some bees wax. Maybe some essential oils if you want to add a little personal touch.
Use a double boiler to warm the mineral oil to the point it will dissolve the wax. Stir it up to mix the two and your done.
One ounce of wax to a 12 ounce bottle makes a thin gel. I prefer around 2 ounces of wax. After the mix cools it is a thick gel. Now you have a long lasting fix for lubricating wooden moving parts and a very good water resistant polish for all your furniture.
Put on thick where you have moving parts, then polish the whole chair and any thing else that needs it.
Works very well on wooden butcher blocks as well. All the ingredients are food safe.