Sometimes, it is the router's responsibility to cool things. I once saw a massive CNC vertical router made by ekstrom carlson which actually had a liquid cooled head. Shortly thereafter, everyone went to using gantry routers and these dinosaurs now sit in the back of used dealers buildings and will be there 20 years from now.
What your talking about is cooling and chip load. You see this in both metalworking and woodworking and routers are nortorious for violating this rule. In short, as the chip leaves the item being worked on, it also removes heat from both the cutter and the item. The heat actually can sacrafice the chip. When I work on my metal lathe, I have metal chips comming off so hot that they are turning blue and I have to wear long sleaves as the chips landing on my forarms leave tiny burn marks. The same applies for woodworking. By maintaining your chip load, you prevent burning of both your wood and your cutters. What happens when you slow down or stop when routing maple and cheery? You get burns and smoke.
What is more important is that you are killing your expensive cutters while doing this. This is one reason router cutters wear out faster than shaper cutters. Most router applications are not run nearly fast enough to maintain the correct chip load and prolong the cutter's life. I dont mean RPM here... rather, the feed rate.
But if this is the source of your collet warming up excessively, what is happening to your cutter's tip? That tip should be smokin' hot and begin to turn blue and carbarize some sawdust. Do you have wood burning as well?
What does your CNC manual tell you about feeds and speeds for the various bits and materials and depth of cut?
Had the dog not stopped to go to the bathroom, he would have caught the rabbit.