Welcome to the site !!
Most likely it is the fact of riding trails .... the cooling system is cooled by airflow, which means air has to flow over the radiators to actually cool the bike ..... the slower the bike goes , the less the airflow , the less cooling effect ...... basically the R was really made for track use where it is going at a fairly consistent speed , since it has no overflow system (overflow tank like you vehicle or designated trail bikes) the temp goes up from the slower riding , the cap blows it off , and out it goes with no ability to return ....
What you could do is replace the radiator cap with a higher BAR cap , the stock would most likely be a 1.1 bar , change it to a 1.6 or 1.8 and the bike will get hotter without overheating/puking the coolant out .... it won't stop the issue but it will help , the other options if you are going to trail ride the bike frequently is to add a overflow bottle/tank , it will allow the coolant to be sucked back in and you will lose maybe 1/10th of what you loose currently , you can get aftermarket ones most any bike shop (or here) .........
You could also purchase a aftermarket Water Pump , it will increase flow ...... but unfortunately , there is not way to really prevent the overflow issue from happening , just reducing it somewhat ...... Basically , trail riding is generally slower than track riding , and the bike will puke coolant , there is no real way around it unless he is riding on fast flowing open trails , but when he gets to walking pace in spots , the bike is going to get hotter , if there is not higher speed runs right after that , it will either take much longer to cool down , or continue to get hotter or stay hot ....