Is it necessary to change the water when washing the feet if its color changes due to dirt, and did the Messenger of Allah, may Allah bless him and grant him peace, change the water for that same reason?

When washing the feet, it is obligatory to make sure that absolute water covers them completely in the first wash. It is then recommended to wash them twice more, and this constitutes the complete washing. As for the sufficient washing, it is done by washing the limb once.

Therefore, if the person performing ablution pours water over the foot and covers it with water, it does not matter if the water changes [color or state] while passing over the foot or dripping from it. It is not a condition for water to drip from the limb at all; rather, it is a condition for water to flow over it.

It is not a condition to clean the foot from non-corporeal dirt so that the water dripping from it is pure. Non-corporeal dirt is not considered a barrier preventing water from reaching the skin. However, corporeal dirt must be removed.