The burner section of the furnace exhausts to the exterior of the rig. That is normal.
You said the furnace cycles on and off, how often? This should happen about 6 times per hour. There is an adjustment on some thermostats to change cycling but it also changes the temperature on/off times. It is the Attenuator. This is on mechanical thermostats.
It is possible that the furnace is shutting down via the over heat sensor. Furnace over heats sensor opens then cools off and restarts. Could be caused by restricted air movement with in the furnace or ducting in the rig.