Praise be to Allah.
What is required in wudoo’ is to wash the face with water, because Allah, may He be exalted, says (interpretation of the meaning):
“O you who believe! When you intend to offer As-Salat (the prayer), wash your faces and your hands (forearms) up to the elbows…”
Washing means pouring water over the part of the body; just wiping it is not sufficient.
Al-Nawawi (may Allah have mercy on him) said: It is stipulated when washing the parts of the body that the water should flow over them; if the water touches but does not flow, then his purification is not valid. Our companions are agreed on this.
End quote from Sharh al-Muhadhdhab, 1/491.
And he (may Allah have mercy on him) said: The Muslims are unanimously agreed that the water that is acceptable in wudoo’ and ghusl is not a specific amount; rather a little or a lot may suffice if the condition of washing is met, which is that the water should flow over the parts of the body.
End quote from Sharh Saheeh Muslim, 2/20
Al-Bahooti (may Allah have mercy on him) said: Proper washing means making the water reach all of the part of the body being washed, so that it flows over it; it is not sufficient just to wipe it.
End quote from Daqaa’iq Ooli an-Nuha, 1/88
To sum up: in order for wudoo’ to be valid, the water must flow over the part of the body in question – apart from the head. But if the water does not flow on the part of the body, and he merely transfers what is left on his hand of water, then that is wiping, in which case his wudoo’ is not valid.
And Allah knows best.