Running shoes in washer/dryer?

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I managed to get ankle deep in mud on my doggy walk today in my running shoes I normally wear for indoor fitness. I'm housesitting/dogsitting and forgot my hiking shoes at home and kind of misjudged a trail I took.... These shoes are old and was planning on getting new ones this month anyways.. however, I wouldn't mind keeping this pair as a backup.

Can I put them in a washer/dryer or will I destroy them? Also, could this damage the expensive washing machine/dryer this house has?

I'd like to use them to work out tomorrow morning.