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The best way to clean white fabric shoes is with an old toothbrush and some form of a cleanser. Murphy's Oil Soap, liquid laundry detergent, Comet, etc. are all good choices. Air dry. Most shoes can be washed in the washing machine, but it will shorten their lifespan, and may cause some of the glue to come loose. If any stains remain after cleaning, the dye will not likely evenly cover the shoes, so you will want to do as thorough of a cleaning as possible.
This tip was right on the head. I was washing my daughter's white fabric tennis shoes in the washer and was having to wash them over at times because they wasn't coming clean. The toothbrush took care of the necessary spot that the washing machine couldn't. Also after washing if you spray some starch on them if helps to keep them cleaner by resisting dirt.