Is it necessary for the water to reach the limb intended for washing in wudu' from the first instance for each of the three washes, or is it sufficient for the water to reach it even if it is during the third wash?

The washing of a limb is not counted until it is completely covered with water. So, if water covers the limb, that counts as the first wash. If the washing is increased beyond three times after the limb has been covered with water, it is disliked, but the ablution remains valid.