There are several other problems - including potential outbreaks of fun pests like fungus gnats and spider mites.
It's much easier to start cuttings later, but there's no harm in starting them now. Just don't overwater them or you'll end up losing them to rot. I let mine teeter on the brink of letting them dry out completely during the winter. If I see any condensation in the inside of the cup, I stop watering.
Oh, and to answer your question: Yes, they will continue to grow (at some reasonable rate or the other) through the winter if you provide them with sunlight and temps above 50º-55º. Note that roots don't like sunlight, so put clear cups inside an opaque cup or wrap the cup with dark paper or foil to keep sun out.