Cloth collision will work with moving objects, but only if you set the Object Type of this object to WorldStatic in the collision settings (and activate Collide with Environment in the cloth skeletal mesh settings).
However, I want my cloth objects to interact with some moving objects and they should not all be WorldStatic. How can I change this?
I could not find such a setting, although I am sure that somewhere in the code it is specified, we just were not given the opportunity to change it for the cloth simulation. If someone tells me where in the code it is written about this for the cloth simulation, I will be glad!