Most running shoes are pretty suitable for machine washing - they are strong and pretty light, so they won't break your household appliance.
They are also strong enough to withstand machine cleaning, but only if you do it right.
Here is advice on how to clean your footwear safely and properly.
Before attempting to wash your footwear, check the special care label inside.
Take out the shoelaces and take off any removable parts if you can.
To protect your footwear from damage, place them inside a special mesh laundry bag or even an old pillowcase.
Set your washing machine to a medium or delicate cycle with pretty cold water.
Don't use bleach or any other harsh chemicals, as they can cause severe discoloration or even structure damage to the footwear.
To cushion the footwear during the washing, you can put a couple of towels inside, or huge sweaters (but not woolen), or basically anything soft.
Place the wet trainers in a properly ventilated area and take them away from direct sunlight.
Stuff them with some crumpled newspaper or thick paper towels to help maintain their shape and get rid of the excess moisture.