You can do the dry cleaning right on the chairs....but taking them off and having them cleaned would be the best bet.
If you find a dry cleaner who knows about antique fabrics you should not have nay trouble with shrinkage.
You could try rubbing them with corn starch and vacuuming well....it will remove a lot of the dirt and it's a "no harm" method....but if they are not cleaned to your liking, remove the covers and have them professionally cleaned.
Linda C

I would take them off the chair seat, head to a good cleaners, and asked that they be dry cleaned and blocked. Blocking will keep the shape square. Without blocking, they'll end up sort of diamond shaped. Should be no extra cost for blocking. Do not have them wet laundered. Most likely the yarn is wool and it'll shrink/fall apart.