We consider a dynamic routing problem where the objective of each user is to obtain flow policy that minimizes its long term cost. The framework differs from other related works which consider collection of static one shot games with dynamic cost function. Instead, we motivate our problem from the two basic facts: i) the path cost may not be exactly known in advance in dynamic environment unlike static; ii) long term solution is important aspect to evaluate rather than obtaining one slot solution. Moreover, this constraint inhibits to apply traditional game theoretic approach to obtain equilibria, rather we discuss that it is not required to obtain equilibria at every slot to "cover" the dynamic environment. In this work we propose an evolutionary game theoretic approach, we intend to learn the optimal strategy exploiting the past experiences (information) instead of cost function. Further, we characterize the dynamic equilibria of the long-term game using evolutionary variational inequalities. The dynamic equilibria so obtained, optimizes the long term cost, however it need not to be an equilibrium for intermediate epochs (games). As a byproduct, this reduces drastically the computation complexity. Under strictly monotone cost function, we prove that the dynamic equilibria are also dynamic evolutionarily stable strategies.