What is the expected amount of DNA shared by first cousins and the people between them in a family tree?
To simplify your problem, let's just consider autosomal DNA. In this case,  your expected amount of shared DNA should be synonymous with genetic consanguinity, which can be conveniently checked from this image. 
Thus, you share approximately 1/8 of DNA(I know saying so is not scientifically strict as there are recombinations and mutations that prevent the DNA to be exactly the same.) with your first cousins. Same is true between first cousins.
Source: Wikipedia image gallery.