LJ60 Captain reported pneumatic source and duct overheat problems resulted in a diversion to make a precautionary landing.

Narrative

While climbing enroute; ZZZ Center held us at FL210 for 3-5 minutes; assumably due to traffic. In that period; we had a dual bleed air master caution light illuminate. We followed the Quick Reference Handbook (QRH) and immediately corrected the situation. Per the QRH; we elected to continue with caution. However; if we received the same master caution; we would divert to our nearest suitable airport as a precaution. We were issued a climb to FL410. Climbing through FL310; we received the dual bleed air light master caution again. We advised ATC that we need to stop the climb at FL320. However; for direction; we had to descend back to FL310. We again began to troubleshoot using the QRH when we also got a DUCT OVHT light. I checked the temperature indicator on the cabin and cockpit duct valves which indicated fully pegged towards the highest setting; although the actual mechanical switches had not been operated yet. This indicated to me that we had either a fire or ignition for a fire to begin in the tail cone section of the airplane.We immediately [requested priority handling] with ZZZ Center and advised them that we need to descend immediately and will be diverting to ZZZ1. We followed the emergency memory items for expedited descent and were cleared to FL240. The First Officer began to troubleshoot using the QRH while I began to manually fly the plane into the descent to ensure we still had control effectiveness; thankfully; we did. Descending through roughly 16;000 ft. we had the bleed lights extinguished and about one minute later; the DUCT light extinguished as well. While configuring for the approach; we had noticed a small sort of vibration or rattling in the plane that was concerning. With that; we advised ZZZ1 Tower that we will be evacuating the aircraft on the runway if there was a visible fire; if not; we will evacuate on the first taxiway available. They advised that there was no smoke present; so we made the decision to exit the runway and evacuate immediately upon clearing due to the nature and location of the potential fire.