Rasengan is technically stronger than the Chidori, but the reason they always cancel eachother out is because of how the attacks work. Chidori condenses all of it's damage into one piercing attack. Rasengan grinds against the opponent over time and then explodes. The Rasengan grinds away the Chidori before reaching critical mass and exploding. This means the Rasengan can't break through in time thanks to exploding, and the Chidori can't get through the Rasengan because it's grinding away, not allowing the lightning blade to pierce clean through.
So yeah, Rasengan is stronger than a Chidori, but it will never defeat a Chidori.
Kirin however, is superior to Rasenshuriken, even if the Rasenshuriken has the elemental advantage for one reason: scale. Kirin uses natural lightning and thus has destructive force on the level that a human could never accomplish with their chakra alone. FRS however is just a Rasengan with Naruto's on wind chakra added to it. He's not taking control over a natural tornadoe and bending it to his will. He's simply putting wind in it.
Not to mention the attacks work differently. Kirin has no upward limit (the bigger the storm, the bigger the amount of lightning you can combine to create the Kirin). It can also be created with far less chakra, hits in 1/1000th of a second, and was strong enough to destroy a Susanoo completely aswell as destroy an entire fortress aswell as the mountain beneath it.
If an FRS was thrown at a Kirin, the Kirin would likely still kill the FRS user because of the collateral damage. Either the FRS and Kirin would both blow up on top of the FRS user, or the Kirin would fall before he can even throw the attack.
So yeah. Rasengan>Chidori. Kirin>>FRS.