Understanding the dynamics and the stabilization of flames is important for practical applications and for theoretical research. Here, we investigate the time evolution of a Bunsen flame in a Poiseuille flow from an initially flat profile to a stationary front position by using a G-equation-based kinematic model. The transient flame positions are given by the solution of the G-equation. To find the solutions of the non-linear G-equation, the analytical method of characteristics is employed resulting in analytic expressions in terms of elliptic integrals for the location of the characteristics and for the location of the flame front along the characteristics. By employing the secant method in inverting the implicit relation giving the location of the characteristics, an analytical–numerical description of the transient position of the location of the flame front is derived. The transient Bunsen flame reaches in finite time a stationary position, whose analytical expression recovers the steady solution of the G-equation for a Poiseuille flow. The stationary position is proved to be stable with respect to small perturbations of the flame front. The time required for stabilization is further characterized by employing a detailed analytical investigation.