The whole point behind this thread is that you said that Kirin is better than the Rasenshuriken on a previous thread.
In some aspects, it probably is. For instance, Kirin harnesses atmospheric energy to create an artificial lightning bolt, and therefore, is probably one of the most powerful attacks in the series. The Rasengan on the other hand is an artificially created Tailed Beast Bomb, and is also probably one of the most powerful attacks in the series. Many people like to throw in real world science at this point to support their argument that Kirin is more powerful, but since the laws of physics don't apply in the same manner in the Narutoverse, I cannot make such a determination. Instead, I will just assume that Kirin and the larger variations of the Rasengan have similar power levels.
That being said, the only real difference is their implementation in combat. With that, the Rasengan (and its variants) beat out Kirin. Kirin takes a long time to set up, and seems to be a one-time use attack. The Rasengan can be formed almost instantly, and can be used repeatedly. Therefore, the Rasengan clearly has an advantage. The only possible exception to this is if there is a storm already overhead, and even then, Kirin will likely exhaust the available energy in one or two attacks.
Assuming that Kirin and the Rasenshuriken are used simultaneously, with identical power levels, Kirin would likely be more effective. The main reason for this is that while Kirin, and even most variations of the Rasengan, are extremely focused attacks, the Rasenshuriken dissipates its attack energy over a wider area. This means that Kirin would probably have more power on impact than the Rasenshuriken. This advantage is only increased when taking the speed of the techniques into account. Regardless of the speed or reaction time of the users, it is fairly obvious that Kirin is faster than the Rasenshuriken.
That being said, a regular Rasengan, on the other hand, could be used to counteract Kirin. Kakashi is known for being able to split lightning bolts with the basic Chidori. This is the same Chidori that Sasuke learned during the Chunin Exams; the same Chidori that Kakashi created to replace the Rasengan in his arsenal after failing to combine it with Lighting; and the same Chidori that was clearly not as powerful as the Rasengan during Sasuke and Naruto's little scuffle on the roof. This indicates that with the right amount of chakra and implementation, the Rasengan could likely be used against a lightning bolt, or a lightning bolt-like attack like Kirin. If Naruto focused the same amount of energy as a Rasenshuriken into a basic Rasengan, he could probably counteract pretty much any technique.
Back on topic, you clearly don't believe that a basic fire attack like the Fire Ball Jutsu is more powerful than any form of the Rasengan. The bottom line is that in most cases it isn't. However, with the right amount of power, even a fairly basic attack like the Fire Ball Jutsu could be used effectively against a relatively powerful attack like the Rasenshuriken. For instance, I'm fairly certain that any fire attack used by Madara could probably crush any form of Rasengan that Konohamaru could come up with. I even think that Naruto, with a Rasenshuriken, would likely struggle against a fire-based attack from Madara.