It is also irrelevant if the airline is the same or not.

In your specific case, it is correct that you must connect in LIM. Also you should check your connection time. I booked this specific routing yesterday and assumed that 60 minutes would be enough time but Lan required 90 minutes and so it means waiting around LIM airport for an extra 4 hours to get the next plane out to CUZ. In addition, they weren't even sure if you could check the luggage all the way through. Does anyone know if you have to pick up your luggage in LIM and check it in again for CUZ.