I don't have a disposal, my kitchen sink fills when my dishwasher empties, why?
It sounds like the pipes below the kitchen / washer are starting to plug up. It would be a good time to run a snake down the line or have a plumber do it. It will get worse once this has started, if you have trees and a older house roots may be making there way into the drain lines and a cutter tip on a drain auger may be required. I mention this because I have owned several older houses that had this problem, one I only had to clean out in the spring the other I had to replace the old concrete drain pipe.