We investigate the translocation of a single-stranded DNA through a pore which fluctuates between two conformations, using coupled master equations. The probability density function of the first passage times of the translocation process is calculated, displaying a triple-, double-, or monopeaked behavior, depending on the interconversion rates between the conformations, the applied electric field, and the initial conditions. The cumulative probability function of the first passage times, in a field-free environment, is shown to have two regimes, characterized by fast and slow timescales. An analytical expression for the mean first passage time of the translocation process is derived, and provides, in addition to the interconversion rates, an extensive characterization of the translocation process. Relationships to experimental observations are discussed.