I turn the chairs upside down and paint everything, even under the chair. If the wood will expand and shrink still, it will do it less if all is painted.
I paint 3 thin layers, and then I have a nearly perfect match with our dining table. The thinner the better, it dries easily and it won't leave dark 'teardrops' of paint.
At this point I am going to cover the bottom with cotton sheeting. Pre-iron the sheeting because cotton shrinking is strong enough to pull your chair apart. I use a small nailgun to secure the cotton over the bottom. Be careful with this tool. I have a very light weight version, which does the trick perfectly well. But if you have something bigger and electric, I would advise you to google "nail gun accident" and you will see that this is not a toy.First place the sheeting over the bottom, secure the middle of each side and then work your way to the corners, folding the cotton as you go.
You don't have to do this very tightly. I always use a hammer to ram the nails in a little further. I'd hate it if my guest would hurt themselves sitting down at my table.
The last step can be seen here.