No, you should not have cold air coming in under your warm roof.
Your cold roof has internal and between rafter insulation and outside airflow above. The outer layers will be open to outside air but inside will be nice and warm.
Your warm roof will have insulation on the roof deck, so the whole thing is warm. It needs no ventilation. Again, no outside air should get in at all.
On the interface you will need continuous insulation, while still allowing the airflow over the outside of the cold roof.
Perhaps post a drawing.