On a physical/scientific level, all temperature is movement (usually on a very small scale, like from molecules and atoms).
A Fire Elemental/Archon has a finite temperature, which means there is a higher temperature. To be able to move in the way a Elemental does (he doesn't need a "fuel line" to move), something must keep him together. If something manages to "add" heat to the elementals temperature it is possible that this bonding force is overwhelmed, ripping the Elemental apart. The rips may be filled with other fire, but this fire is not part of the Elemental, and it works as well as cutting a human in pieces and trying to "repair" him by putting flesh in the resulting holes (which means it can work, but not on a "big" scale.
A Ice Elemental/Archon has a low temperature. But if he was at 0 K, he couldn't move (especially not in the way he probably does. Quantum Effects might still allow a little bit of movement, since he can't stay at 0 K anyway.). There is force at work that keeps him cold, but also not too cold. If you can get him cold enough, the force that stabilizes his temperature can fail. It's as if you would cut off the blood supply of living cells. If done shortly enough, it can be repaired, but if it's to long, the cells die. If enough cells die, the Ice Archon dies...