This paper reports the survival probability of unstable quantum states when a charged particle, controlled by the external time-varying field, tunnels through a moving double barrier with a potential well inside. An analytical solution is derived to the non-stationary Schrödinger equation for wave packet transmission through a double barrier, including analytical expressions for wave functions and transmission-reflection amplitudes in the rest frame of the barrier. Solutions corresponding to bound states and scattering states for the cases of stationary and moving barriers are jointly investigated. It is shown that the external time-varying field can be used both to increase and to decrease the survival probability by many orders of magnitude as compared to the stationary barrier. The survival probability at long times is represented by an asymptotic series whose coefficients depend on the external alternating field. The contribution of the bound states to this probability can be varied within a wide range by changing the distance between the wave packet localization point and the center of the double-barrier structure.