Understanding the role of network structure in the evolution of cooperation is a key research goal at the intersection between physics and biology. Recent studies have particularly focused on multiplex networks given that multiple social domains are interrelated and cannot be represented by single-layer networks. However, the role of network multiplexity is not fully understood when combined with another important network characteristic: network dynamics. In the present study, we investigated evolutionary prisoner’s dilemma games played on dynamic two-layer multiplex networks in which the payoff combined across the two layers determined strategy evolution. In addition, we introduced network dynamics where agents can sever links with defecting neighbors and construct new links. Our simulation showed that link updating enhances cooperation but the resultant states are far from those of full cooperation. This modest enhancement in cooperation was related to symmetry breaking whereby the cooperation frequency in one layer disproportionately increased while that in the other layer remained the same or even diminished. However, this broken symmetry disappeared with sufficiently fast link updating. Our results show that the introduction of network dynamics enhances cooperation in the prisoner’s dilemma as previously reported, but this enhancement is accompanied by significant asymmetry once network multiplexity is considered.