This work introduces a simple mean-field network game that captures some of the key dynamic features of crowd and pedestrian flows in multi-level building evacuations. It considers a route choice by finite number of strategic agents. Each agent state is represented by a simple first order dynamical system. Each agent measures its local congestion measure based on its location in the building. Including the local congestion term and its evolution along the path causes a sort of dispersion of the flow: the agents will try to avoid high density areas in order to reduce their overall walking costs and queuing cost at the exits. Each agent will move to one the closest exits that is safer and that has less congestion through the path. The dynamics of lower layer floors are influenced not only by the agents in that floor but also by the incoming flow from the upper layers through the stairs leading to interacting flows between the floors. Numerics and simulations are carried out to illustrate mean-field equilibria of the evacuation process.