Praise be to Allah.
The outside of a thick beard should be washed, but it is not a must to wash the inside or the skin underneath. However, it is encouraged to run one's fingers through the beard. Al-Nawawi (may Allaah have mercy on him) said: The scholars agree unanimously that a thick beard should be washed, but that the inner layers do not need to be washed, nor does the skin underneath. This is the opinion of the majority of scholars among the Sahaabah, Taabieen and those who came after them. Ibn Rushd said: This is an issue in which I know of no difference of opinion.
However, if the beard is thin and the skin underneath is visible, then it should be washed thoroughly when performing wudoo.