In this paper, we propose an evolutionary game-theoretic framework for hybrid additive white Gaussian noise multiple access channels. We consider a communication system consisting of multiple users and multiple receivers, where each user chooses a rate and splits it over the receivers. Users have coupled constraints determined by the capacity regions. We show the existence of Nash equilibrium under general conditions and characterize the equilibria of the static game. Building upon the static game, we formulate a system of hybrid evolutionary game dynamics using G-function dynamics and Smith dynamics on rate control and channel selection, respectively. We show that the evolutionary hybrid multiple access game has an equilibrium and illustrate these dynamics with numerical examples.