In this paper, we study the asymptotics of the degree sequence of permutation graphs associated with a sequence of random permutations. The limiting finite-dimensional distributions of the degree proportions are established using results from graph and permutation limit theories. In particular, we show that for a uniform random permutation, the joint distribution of the degree proportions of the vertices labeled ⌈nr1⌉,⌈nr2⌉,…,⌈nrs⌉ in the associated permutation graph converges to independent random variables D(r1), D(r2),…, D(rs), where D(ri)∼Unif(ri,1−ri), for ri ∈ [0,1] and i ∈ {1,2,…,s}. Moreover, the degree proportion of the mid-vertex (the vertex labeled n/2) has a central limit theorem, and the minimum degree converges to a Rayleigh distribution after an appropriate scaling. Finally, the asymptotic finite-dimensional distributions of the permutation graph associated with a Mallows random permutation is determined, and interesting phase transitions are observed. Our results extend to other nonuniform measures on permutations as well.