Yes, this probably happens to all of us who have steel rotors, just some people have more tolerance to dust on their wheels 😇
Try driving hundred meters/yards immediately after you finish washing, before drying, and braking firmly every now and then. This wipes most of the moisture off the rotors reducing the rust like residue that then comes off when you start using the brakes after wash. Doesn’t eliminate the problem, but reduces the brown dust after first drive quite significantly.
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