We study reaction fronts inside a two-dimensional domain subject to a Poiseuille flow. We focus on a cubic reaction–diffusion system with two chemicals having different diffusion coefficients. We solve numerically the system of equations for different values of the domain width finding transitions between traveling steady states. These transitions are also observed by changing the average velocity of the external Poiseuille flow. The flat front solutions are obtained using the equations in a one-dimensional region, and extending them to two-dimensions. These solutions result in either stable or unstable fronts. We carry out a linear stability analysis for flat fronts obtaining the corresponding growth rates of small perturbations. The application of a Poiseuille flow in the same direction of the propagating front gives rise to stable symmetric fronts, whereas in the opposite direction allows the formation of stable symmetric and asymmetric fronts. For strong enough flow velocities or wide widths, the fronts become oscillatory. Increasing the driving parameters results in intermittent bursts in the oscillations.