Assuming that a missile is able to track, lock and effectively guide itself to another missile for sake of argument, would say an AIM 9X or AIM120 be able to shoot down its respective counterpart?
Note the AIM 9X and AIM 120 are just examples, I'm talking about in general. Air launched air to air missiles are the scope of the question. So anti-ballistic missiles, cruise missiles etc are out of scope.
The reason I ask is because active protection systems against kinetic energy penetrators, especially depleted uranium rounds, face significant challenges in achieving hardkill. An APS system isn't going to do much other than breaking it up a bit or causing the incoming round to tumble or wobble. Either way an APFSDS round is smashing against hull armor and punching a nasty gash at the very least. Speed is one of the issues that make APS systems ineffective compared to stopping HEAT.
Missiles on the other hand are launched way higher in the sky, with a high initial velocity and a powerful rocket motor. While the motor quits eventually, there's still a lot of kinetic energy. However, missiles are more beholden to the laws of aerodynamics and loss of kinetic energy. I'm not entirely sure if blast frag or a continuous rod warhead could expand fast enough to cut up a missile. Or if there was non catastrophic damage, would the damaged missile body and fins allow a missile to still limp its way to a target for a successful kill, or would the missile bleed kinetic energy like crazy eventually falling out or self detonating.