The key to identifying your problem could well be connected to the point that you made when you said the situation happens “when you run at high intensity”.
Your body is talking to you, let’s try and understand what it’s saying. Keep in mind you have a network of high performance muscles running through the lower right hip area. The tightness you feel could well be that your hip flexor muscle is tightening up.
When you up your speed and performance your body is trying to tell you that something is not right and this results in a feeling of tightness which can lead onto tendon or muscle pain which if unchecked may well cause cramp.
In short, you’re not bringing your leg forward quickly enough each time you increase your speed and that’s causing drag on your hip flexor muscle, overstretching it.
Listen to your body and react and go with it, bring your leg forward more quickly and the tightness will decrease. Next time you run, focus on this and see if you get a good result.
Let us know how things work out.