It has dedicated tracking modes for them (the mount tracks in both RA and Dec).
For lunar, here's a quote from the manual:
This submenu allows you to select specific features of the Moon such as craters.
– Select feature allows you to choose the specific feature you wish to observe. Available features are: Crater, Albedo feature, Catena, Dorsum, Lacus, Landing site, Mare, Mons, Promontorium, Rima, Rupes, Satellite feature, Sinus, Vallis. The Landing site features are further divided into sections corresponding to the different missions.
– Coordinates allows you to specify selenographic coordinates to aim to (lunar longitude and latitude).
– Terminator aims at the center of the terminator.
– Sync on feature is an additional function that allows you to obtain additional accuracy on the moon. To use this function proceed as follows. Select a small, recognizable feature on the Moon and slew to it. Then center the feature accurately. Now select Sync on feature . From now on, the position of all features you will select on the Moon will be corrected taking the feature you synchronized on as a reference. If the synchronization is not satisfactory, choose Clear sync to remove the correction. Note that this type of synchronization works only on the moon, and is completely unrelated to the alignment model.
For comets and asteroids, similarly, once you upload recent data from the MPC, you can track them.
In a similar manner, it also tracks artificial satellites, that move much faster but that's out of the scope of the question.