Aims. Most studies of the dynamics of main-belt objects describing the evolution of the bodies in inner Solar System have been carried with models that include weak nongravitational forces, such as the Yarkovsky and YORP effects. Only about 19 objects exhibit cometary-type activity, with sublimation being the principal mechanism. This paper presents a study of the influence of cometary- type nongravitational forces on the dynamics of main-belt comets, and the possible paths and timescales of evolution into other Solar System regions. Methods. We used the standard Marsden model for cometary-type activity. This model was designed for elongated orbits and the continuous ejection of mass, while the main-belt comets exhibit a different mode of activity. For this reason, we propose a simple model of nongravitational force activation that is consistent with observations. The dynamical evolution of objects was studied using a fifteenth-order RADAU integrator implemented in the REBOUND package. Results. The paper presents the dynamical routes of main-belt comets in the inner Solar System when cometary-type nongravitational forces are included in calculations. The forces significantly shorten the time of transition to other regions compared to the Yarkovsky effect, shortening this time to as little as a few thousand years depending on the frequency of the activity and the A1, A2, and A3 Marsden constants. There is a large probability of a transition to near-Earth object(NEO)-type orbits for bodies with A2 < 0, which means that cometary-type nongravitational forces can be a non-negligible mechanism increasing the number of bodies in that population. The forces can also deliver active main-belt objects into mean motion resonances but can equally eject bodies into outer planetary regions on far shorter timescales than the Yarkovsky and YORP effects. Cometary-type nongravitational effects should be included in dynamical studies of individual sublimating active asteroids.